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Theorem sup2 12201

Description: A nonempty, bounded-above set of reals has a supremum. Stronger version of completeness axiom (it has a slightly weaker antecedent). (Contributed by NM, 19-Jan-1997.)

**Assertion**
Ref	Expression
sup2	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

Distinct variable group: 𝑥,𝑦,𝑧,𝐴

**Proof of Theorem sup2**
Step	Hyp	Ref	Expression
1		peano2re 11418	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ ℝ → (𝑥 + 1) ∈ ℝ)
2	1	adantr 480	. . . . . . . . . . 11 ⊢ ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → (𝑥 + 1) ∈ ℝ)
3	2	a1i 11	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → (𝑥 + 1) ∈ ℝ))
4		ssel 3973	. . . . . . . . . . . . . . . 16 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ))
5		ltp1 12085	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑥 ∈ ℝ → 𝑥 < (𝑥 + 1))
6	1	ancli 548	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (𝑥 ∈ ℝ → (𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ))
7		lttr 11321	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ) → ((𝑦 < 𝑥 ∧ 𝑥 < (𝑥 + 1)) → 𝑦 < (𝑥 + 1)))
8	7	3expb 1118	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑦 ∈ ℝ ∧ (𝑥 ∈ ℝ ∧ (𝑥 + 1) ∈ ℝ)) → ((𝑦 < 𝑥 ∧ 𝑥 < (𝑥 + 1)) → 𝑦 < (𝑥 + 1)))
9	6, 8	sylan2 592	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦 < 𝑥 ∧ 𝑥 < (𝑥 + 1)) → 𝑦 < (𝑥 + 1)))
10	5, 9	sylan2i 605	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦 < 𝑥 ∧ 𝑥 ∈ ℝ) → 𝑦 < (𝑥 + 1)))
11	10	exp4b 430	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝑦 ∈ ℝ → (𝑥 ∈ ℝ → (𝑦 < 𝑥 → (𝑥 ∈ ℝ → 𝑦 < (𝑥 + 1)))))
12	11	com34 91	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 ∈ ℝ → (𝑥 ∈ ℝ → (𝑥 ∈ ℝ → (𝑦 < 𝑥 → 𝑦 < (𝑥 + 1)))))
13	12	pm2.43d 53	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ ℝ → (𝑥 ∈ ℝ → (𝑦 < 𝑥 → 𝑦 < (𝑥 + 1))))
14	13	imp 406	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 < 𝑥 → 𝑦 < (𝑥 + 1)))
15		breq1 5151	. . . . . . . . . . . . . . . . . . . 20 ⊢ (𝑦 = 𝑥 → (𝑦 < (𝑥 + 1) ↔ 𝑥 < (𝑥 + 1)))
16	5, 15	syl5ibrcom 246	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 ∈ ℝ → (𝑦 = 𝑥 → 𝑦 < (𝑥 + 1)))
17	16	adantl 481	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 = 𝑥 → 𝑦 < (𝑥 + 1)))
18	14, 17	jaod 858	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → 𝑦 < (𝑥 + 1)))
19	18	ex 412	. . . . . . . . . . . . . . . 16 ⊢ (𝑦 ∈ ℝ → (𝑥 ∈ ℝ → ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → 𝑦 < (𝑥 + 1))))
20	4, 19	syl6 35	. . . . . . . . . . . . . . 15 ⊢ (𝐴 ⊆ ℝ → (𝑦 ∈ 𝐴 → (𝑥 ∈ ℝ → ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → 𝑦 < (𝑥 + 1)))))
21	20	com23 86	. . . . . . . . . . . . . 14 ⊢ (𝐴 ⊆ ℝ → (𝑥 ∈ ℝ → (𝑦 ∈ 𝐴 → ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → 𝑦 < (𝑥 + 1)))))
22	21	imp 406	. . . . . . . . . . . . 13 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (𝑦 ∈ 𝐴 → ((𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → 𝑦 < (𝑥 + 1))))
23	22	a2d 29	. . . . . . . . . . . 12 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → ((𝑦 ∈ 𝐴 → (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → (𝑦 ∈ 𝐴 → 𝑦 < (𝑥 + 1))))
24	23	ralimdv2 3160	. . . . . . . . . . 11 ⊢ ((𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ) → (∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → ∀𝑦 ∈ 𝐴 𝑦 < (𝑥 + 1)))
25	24	expimpd 453	. . . . . . . . . 10 ⊢ (𝐴 ⊆ ℝ → ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∀𝑦 ∈ 𝐴 𝑦 < (𝑥 + 1)))
26	3, 25	jcad 512	. . . . . . . . 9 ⊢ (𝐴 ⊆ ℝ → ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ((𝑥 + 1) ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < (𝑥 + 1))))
27		ovex 7453	. . . . . . . . . 10 ⊢ (𝑥 + 1) ∈ V
28		eleq1 2817	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥 + 1) → (𝑧 ∈ ℝ ↔ (𝑥 + 1) ∈ ℝ))
29		breq2 5152	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑥 + 1) → (𝑦 < 𝑧 ↔ 𝑦 < (𝑥 + 1)))
30	29	ralbidv 3174	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑥 + 1) → (∀𝑦 ∈ 𝐴 𝑦 < 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑦 < (𝑥 + 1)))
31	28, 30	anbi12d 631	. . . . . . . . . 10 ⊢ (𝑧 = (𝑥 + 1) → ((𝑧 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑧) ↔ ((𝑥 + 1) ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < (𝑥 + 1))))
32	27, 31	spcev 3593	. . . . . . . . 9 ⊢ (((𝑥 + 1) ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < (𝑥 + 1)) → ∃𝑧(𝑧 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑧))
33	26, 32	syl6 35	. . . . . . . 8 ⊢ (𝐴 ⊆ ℝ → ((𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑧(𝑧 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑧)))
34	33	exlimdv 1929	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → (∃𝑥(𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑧(𝑧 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑧)))
35		eleq1 2817	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (𝑧 ∈ ℝ ↔ 𝑥 ∈ ℝ))
36		breq2 5152	. . . . . . . . . 10 ⊢ (𝑧 = 𝑥 → (𝑦 < 𝑧 ↔ 𝑦 < 𝑥))
37	36	ralbidv 3174	. . . . . . . . 9 ⊢ (𝑧 = 𝑥 → (∀𝑦 ∈ 𝐴 𝑦 < 𝑧 ↔ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
38	35, 37	anbi12d 631	. . . . . . . 8 ⊢ (𝑧 = 𝑥 → ((𝑧 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑧) ↔ (𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥)))
39	38	cbvexvw 2033	. . . . . . 7 ⊢ (∃𝑧(𝑧 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑧) ↔ ∃𝑥(𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
40	34, 39	imbitrdi 250	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → (∃𝑥(𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑥(𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥)))
41		df-rex 3068	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥) ↔ ∃𝑥(𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
42		df-rex 3068	. . . . . 6 ⊢ (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥 ↔ ∃𝑥(𝑥 ∈ ℝ ∧ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
43	40, 41, 42	3imtr4g 296	. . . . 5 ⊢ (𝐴 ⊆ ℝ → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
44	43	adantr 480	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥) → ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
45	44	imdistani 568	. . 3 ⊢ (((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
46		df-3an 1087	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) ↔ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)))
47		df-3an 1087	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥) ↔ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
48	45, 46, 47	3imtr4i 292	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → (𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥))
49		axsup 11320	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 𝑦 < 𝑥) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))
50	48, 49	syl 17	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃𝑥 ∈ ℝ ∀𝑦 ∈ 𝐴 (𝑦 < 𝑥 ∨ 𝑦 = 𝑥)) → ∃𝑥 ∈ ℝ (∀𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀𝑦 ∈ ℝ (𝑦 < 𝑥 → ∃𝑧 ∈ 𝐴 𝑦 < 𝑧)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ wo 846 ∧ w3a 1085 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 ∀wral 3058 ∃wrex 3067 ⊆ wss 3947 ∅c0 4323 class class class wbr 5148 (class class class)co 7420 ℝcr 11138 1c1 11140 + caddc 11142 < clt 11279

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 ax-resscn 11196 ax-1cn 11197 ax-icn 11198 ax-addcl 11199 ax-addrcl 11200 ax-mulcl 11201 ax-mulrcl 11202 ax-mulcom 11203 ax-addass 11204 ax-mulass 11205 ax-distr 11206 ax-i2m1 11207 ax-1ne0 11208 ax-1rid 11209 ax-rnegex 11210 ax-rrecex 11211 ax-cnre 11212 ax-pre-lttri 11213 ax-pre-lttrn 11214 ax-pre-ltadd 11215 ax-pre-mulgt0 11216 ax-pre-sup 11217

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-nel 3044 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5576 df-po 5590 df-so 5591 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-riota 7376 df-ov 7423 df-oprab 7424 df-mpo 7425 df-er 8725 df-en 8965 df-dom 8966 df-sdom 8967 df-pnf 11281 df-mnf 11282 df-xr 11283 df-ltxr 11284 df-le 11285 df-sub 11477 df-neg 11478

This theorem is referenced by: sup3 12202 nnunb 12499

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