Theorem nocvxmin 27724

Description: Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. Lemma 0 of [Alling] p. 185. (Contributed by Scott Fenton, 30-Jun-2011.)

Assertion

Ref	Expression
nocvxmin	⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃!𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))

Distinct variable group:   𝑤,𝐴,𝑥,𝑦,𝑧

Proof of Theorem nocvxmin

Dummy variable 𝑡 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		imassrn 6074	. . . . . 6 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 27721	. . . . . 6 ⊢ ran bday = On
3	1, 2	sseqtri 4016	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ On
4		bdaydm 27720	. . . . . . . . . . 11 ⊢ dom bday = No
5	4	sseq2i 4009	. . . . . . . . . 10 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
6		bdayfun 27718	. . . . . . . . . . 11 ⊢ Fun bday
7		funfvima2 7243	. . . . . . . . . . 11 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
8	6, 7	mpan 689	. . . . . . . . . 10 ⊢ (𝐴 ⊆ dom bday → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
9	5, 8	sylbir 234	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
10		elex2 2808	. . . . . . . . 9 ⊢ (( bday ‘𝑥) ∈ ( bday “ 𝐴) → ∃𝑤 𝑤 ∈ ( bday “ 𝐴))
11	9, 10	syl6 35	. . . . . . . 8 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ∃𝑤 𝑤 ∈ ( bday “ 𝐴)))
12	11	exlimdv 1929	. . . . . . 7 ⊢ (𝐴 ⊆ No → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑤 𝑤 ∈ ( bday “ 𝐴)))
13		n0 4347	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
14		n0 4347	. . . . . . 7 ⊢ (( bday “ 𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( bday “ 𝐴))
15	12, 13, 14	3imtr4g 296	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝐴 ≠ ∅ → ( bday “ 𝐴) ≠ ∅))
16	15	impcom 407	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ( bday “ 𝐴) ≠ ∅)
17		onint 7793	. . . . 5 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
18	3, 16, 17	sylancr 586	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
19		bdayfn 27719	. . . . . 6 ⊢ bday Fn No
20		fvelimab 6971	. . . . . 6 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
21	19, 20	mpan 689	. . . . 5 ⊢ (𝐴 ⊆ No → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
22	21	adantl 481	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
23	18, 22	mpbid 231	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))
24	23	3adant3 1130	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))
25		ssel 3973	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → 𝑤 ∈ No ))
26		ssel 3973	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑡 ∈ 𝐴 → 𝑡 ∈ No ))
27	25, 26	anim12d 608	. . . . . . . 8 ⊢ (𝐴 ⊆ No → ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑤 ∈ No ∧ 𝑡 ∈ No )))
28	27	imp 406	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ (𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴)) → (𝑤 ∈ No ∧ 𝑡 ∈ No ))
29	28	ad2ant2r 746	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ∧ (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)))) → (𝑤 ∈ No ∧ 𝑡 ∈ No ))
30		nocvxminlem 27723	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ∧ (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴))) → ¬ 𝑤 <s 𝑡))
31	30	imp 406	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ∧ (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)))) → ¬ 𝑤 <s 𝑡)
32		ancom 460	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴))
33		ancom 460	. . . . . . . . 9 ⊢ ((( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)) ↔ (( bday ‘𝑡) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
34	32, 33	anbi12i 627	. . . . . . . 8 ⊢ (((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ∧ (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴))) ↔ ((𝑡 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (( bday ‘𝑡) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑤) = ∩ ( bday “ 𝐴))))
35		nocvxminlem 27723	. . . . . . . 8 ⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑡 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴) ∧ (( bday ‘𝑡) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑤) = ∩ ( bday “ 𝐴))) → ¬ 𝑡 <s 𝑤))
36	34, 35	biimtrid 241	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → (((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ∧ (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴))) → ¬ 𝑡 <s 𝑤))
37	36	imp 406	. . . . . 6 ⊢ (((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ∧ (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)))) → ¬ 𝑡 <s 𝑤)
38		slttrieq2 27696	. . . . . . 7 ⊢ ((𝑤 ∈ No ∧ 𝑡 ∈ No ) → (𝑤 = 𝑡 ↔ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)))
39	38	biimpar 477	. . . . . 6 ⊢ (((𝑤 ∈ No ∧ 𝑡 ∈ No ) ∧ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)) → 𝑤 = 𝑡)
40	29, 31, 37, 39	syl12anc 836	. . . . 5 ⊢ (((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) ∧ ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ∧ (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)))) → 𝑤 = 𝑡)
41	40	exp32 420	. . . 4 ⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → ((( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)) → 𝑤 = 𝑡)))
42	41	ralrimivv 3195	. . 3 ⊢ ((𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∀𝑤 ∈ 𝐴 ∀𝑡 ∈ 𝐴 ((( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)) → 𝑤 = 𝑡))
43	42	3adant1 1128	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∀𝑤 ∈ 𝐴 ∀𝑡 ∈ 𝐴 ((( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)) → 𝑤 = 𝑡))
44		fveqeq2 6906	. . 3 ⊢ (𝑤 = 𝑡 → (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ↔ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)))
45	44	reu4 3726	. 2 ⊢ (∃!𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴) ↔ (∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ∀𝑤 ∈ 𝐴 ∀𝑡 ∈ 𝐴 ((( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)) → 𝑤 = 𝑡)))
46	24, 43, 45	sylanbrc 582	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃!𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 395   ∧ w3a 1085   = wceq 1534  ∃wex 1774   ∈ wcel 2099   ≠ wne 2937  ∀wral 3058  ∃wrex 3067  ∃!wreu 3371   ⊆ wss 3947  ∅c0 4323  ∩ cint 4949   class class class wbr 5148  dom cdm 5678  ran crn 5679   “ cima 5681  Oncon0 6369  Fun wfun 6542   Fn wfn 6543  ‘cfv 6548   No csur 27586   <s cslt 27587   bday cbday 27588

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-rep 5285  ax-sep 5299  ax-nul 5306  ax-pr 5429  ax-un 7740

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-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-pss 3966  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-tp 4634  df-op 4636  df-uni 4909  df-int 4950  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5576  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  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-ord 6372  df-on 6373  df-suc 6375  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-1o 8487  df-2o 8488  df-no 27589  df-slt 27590  df-bday 27591

This theorem is referenced by:  conway  27745

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