supxrgtmnf - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > supxrgtmnf		Structured version Visualization version GIF version

Theorem supxrgtmnf 13340

Description: The supremum of a nonempty set of reals is greater than minus infinity. (Contributed by NM, 2-Feb-2006.)

**Assertion**
Ref	Expression
supxrgtmnf	⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ^*, < ))

**Proof of Theorem supxrgtmnf**
Step	Hyp	Ref	Expression
1		supxrbnd 13339	. . . . . 6 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ sup(𝐴, ℝ^, < ) < +∞) → sup(𝐴, ℝ^, < ) ∈ ℝ)
2	1	3expia 1119	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) < +∞ → sup(𝐴, ℝ^, < ) ∈ ℝ))
3	2	con3d 152	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (¬ sup(𝐴, ℝ^, < ) ∈ ℝ → ¬ sup(𝐴, ℝ^, < ) < +∞))
4		ressxr 11288	. . . . . . . 8 ⊢ ℝ ⊆ ℝ^*
5		sstr 3988	. . . . . . . 8 ⊢ ((𝐴 ⊆ ℝ ∧ ℝ ⊆ ℝ^) → 𝐴 ⊆ ℝ^)
6	4, 5	mpan2 690	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ → 𝐴 ⊆ ℝ^*)
7		supxrcl 13326	. . . . . . 7 ⊢ (𝐴 ⊆ ℝ^* → sup(𝐴, ℝ^, < ) ∈ ℝ^)
8	6, 7	syl 17	. . . . . 6 ⊢ (𝐴 ⊆ ℝ → sup(𝐴, ℝ^, < ) ∈ ℝ^)
9	8	adantr 480	. . . . 5 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → sup(𝐴, ℝ^, < ) ∈ ℝ^)
10		nltpnft 13175	. . . . 5 ⊢ (sup(𝐴, ℝ^, < ) ∈ ℝ^ → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
11	9, 10	syl 17	. . . 4 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) = +∞ ↔ ¬ sup(𝐴, ℝ^, < ) < +∞))
12	3, 11	sylibrd 259	. . 3 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (¬ sup(𝐴, ℝ^, < ) ∈ ℝ → sup(𝐴, ℝ^, < ) = +∞))
13	12	orrd 862	. 2 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → (sup(𝐴, ℝ^, < ) ∈ ℝ ∨ sup(𝐴, ℝ^, < ) = +∞))
14		mnfltxr 13139	. 2 ⊢ ((sup(𝐴, ℝ^, < ) ∈ ℝ ∨ sup(𝐴, ℝ^, < ) = +∞) → -∞ < sup(𝐴, ℝ^*, < ))
15	13, 14	syl 17	1 ⊢ ((𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅) → -∞ < sup(𝐴, ℝ^*, < ))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 846 = wceq 1534 ∈ wcel 2099 ≠ wne 2937 ⊆ wss 3947 ∅c0 4323 class class class wbr 5148 supcsup 9463 ℝcr 11137 +∞cpnf 11275 -∞cmnf 11276 ℝ^*cxr 11277 < clt 11278

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-cnex 11194 ax-resscn 11195 ax-1cn 11196 ax-icn 11197 ax-addcl 11198 ax-addrcl 11199 ax-mulcl 11200 ax-mulrcl 11201 ax-mulcom 11202 ax-addass 11203 ax-mulass 11204 ax-distr 11205 ax-i2m1 11206 ax-1ne0 11207 ax-1rid 11208 ax-rnegex 11209 ax-rrecex 11210 ax-cnre 11211 ax-pre-lttri 11212 ax-pre-lttrn 11213 ax-pre-ltadd 11214 ax-pre-mulgt0 11215 ax-pre-sup 11216

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-rmo 3373 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 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-sup 9465 df-pnf 11280 df-mnf 11281 df-xr 11282 df-ltxr 11283 df-le 11284 df-sub 11476 df-neg 11477

This theorem is referenced by: supxrre1 13341 ovolunlem1a 25424 suplesup 44721

W3C validator

	
		OSZAR »