Theorem nocvxmin 27516

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 6069	. . . . . 6 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 27513	. . . . . 6 ⊢ ran bday = On
3	1, 2	sseqtri 4017	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ On
4		bdaydm 27512	. . . . . . . . . . 11 ⊢ dom bday = No
5	4	sseq2i 4010	. . . . . . . . . 10 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
6		bdayfun 27510	. . . . . . . . . . 11 ⊢ Fun bday
7		funfvima2 7234	. . . . . . . . . . 11 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
8	6, 7	mpan 686	. . . . . . . . . 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 2810	. . . . . . . . 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 1934	. . . . . . 7 ⊢ (𝐴 ⊆ No → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑤 𝑤 ∈ ( bday “ 𝐴)))
13		n0 4345	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
14		n0 4345	. . . . . . 7 ⊢ (( bday “ 𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( bday “ 𝐴))
15	12, 13, 14	3imtr4g 295	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝐴 ≠ ∅ → ( bday “ 𝐴) ≠ ∅))
16	15	impcom 406	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ( bday “ 𝐴) ≠ ∅)
17		onint 7780	. . . . 5 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
18	3, 16, 17	sylancr 585	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
19		bdayfn 27511	. . . . . 6 ⊢ bday Fn No
20		fvelimab 6963	. . . . . 6 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
21	19, 20	mpan 686	. . . . 5 ⊢ (𝐴 ⊆ No → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
22	21	adantl 480	. . . 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 3974	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → 𝑤 ∈ No ))
26		ssel 3974	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑡 ∈ 𝐴 → 𝑡 ∈ No ))
27	25, 26	anim12d 607	. . . . . . . 8 ⊢ (𝐴 ⊆ No → ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑤 ∈ No ∧ 𝑡 ∈ No )))
28	27	imp 405	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ (𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴)) → (𝑤 ∈ No ∧ 𝑡 ∈ No ))
29	28	ad2ant2r 743	. . . . . 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 27515	. . . . . . 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 405	. . . . . 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 459	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴))
33		ancom 459	. . . . . . . . 9 ⊢ ((( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)) ↔ (( bday ‘𝑡) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
34	32, 33	anbi12i 625	. . . . . . . 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 27515	. . . . . . . 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 405	. . . . . 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 27489	. . . . . . 7 ⊢ ((𝑤 ∈ No ∧ 𝑡 ∈ No ) → (𝑤 = 𝑡 ↔ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)))
39	38	biimpar 476	. . . . . 6 ⊢ (((𝑤 ∈ No ∧ 𝑡 ∈ No ) ∧ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)) → 𝑤 = 𝑡)
40	29, 31, 37, 39	syl12anc 833	. . . . 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 419	. . . 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 3196	. . 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 6899	. . 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 581	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 394   ∧ w3a 1085   = wceq 1539  ∃wex 1779   ∈ wcel 2104   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  ∃!wreu 3372   ⊆ wss 3947  ∅c0 4321  ∩ cint 4949   class class class wbr 5147  dom cdm 5675  ran crn 5676   “ cima 5678  Oncon0 6363  Fun wfun 6536   Fn wfn 6537  ‘cfv 6542   No csur 27379   <s cslt 27380   bday cbday 27381

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-rep 5284  ax-sep 5298  ax-nul 5305  ax-pr 5426  ax-un 7727

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3374  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-tp 4632  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ord 6366  df-on 6367  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-1o 8468  df-2o 8469  df-no 27382  df-slt 27383  df-bday 27384

This theorem is referenced by:  conway  27537