Theorem nocvxmin 33361

Description: Given a nonempty convex class of surreals, there is a unique birthday-minimal element of that class. (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 5907	. . . . . 6 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 33358	. . . . . 6 ⊢ ran bday = On
3	1, 2	sseqtri 3951	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ On
4		bdaydm 33357	. . . . . . . . . . 11 ⊢ dom bday = No
5	4	sseq2i 3944	. . . . . . . . . 10 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
6		bdayfun 33355	. . . . . . . . . . 11 ⊢ Fun bday
7		funfvima2 6971	. . . . . . . . . . 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 238	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
10		elex2 3463	. . . . . . . . 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 4260	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
14		n0 4260	. . . . . . 7 ⊢ (( bday “ 𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( bday “ 𝐴))
15	12, 13, 14	3imtr4g 299	. . . . . 6 ⊢ (𝐴 ⊆ No → (𝐴 ≠ ∅ → ( bday “ 𝐴) ≠ ∅))
16	15	impcom 411	. . . . 5 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ( bday “ 𝐴) ≠ ∅)
17		onint 7490	. . . . 5 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
18	3, 16, 17	sylancr 590	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
19		bdayfn 33356	. . . . . 6 ⊢ bday Fn No
20		fvelimab 6712	. . . . . 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 485	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
23	18, 22	mpbid 235	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))
24	23	3adant3 1129	. 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 3908	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → 𝑤 ∈ No ))
26		ssel 3908	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑡 ∈ 𝐴 → 𝑡 ∈ No ))
27	25, 26	anim12d 611	. . . . . . . 8 ⊢ (𝐴 ⊆ No → ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑤 ∈ No ∧ 𝑡 ∈ No )))
28	27	imp 410	. . . . . . 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 33360	. . . . . . 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 410	. . . . . 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 464	. . . . . . . . 9 ⊢ ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) ↔ (𝑡 ∈ 𝐴 ∧ 𝑤 ∈ 𝐴))
33		ancom 464	. . . . . . . . 9 ⊢ ((( bday ‘𝑤) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)) ↔ (( bday ‘𝑡) = ∩ ( bday “ 𝐴) ∧ ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
34	32, 33	anbi12i 629	. . . . . . . 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 33360	. . . . . . . 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	syl5bi 245	. . . . . . 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 410	. . . . . 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 33342	. . . . . . 7 ⊢ ((𝑤 ∈ No ∧ 𝑡 ∈ No ) → (𝑤 = 𝑡 ↔ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)))
39	38	biimpar 481	. . . . . 6 ⊢ (((𝑤 ∈ No ∧ 𝑡 ∈ No ) ∧ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)) → 𝑤 = 𝑡)
40	29, 31, 37, 39	syl12anc 835	. . . . 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 424	. . . 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 3155	. . 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 1127	. 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 6654	. . 3 ⊢ (𝑤 = 𝑡 → (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ↔ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)))
45	44	reu4 3670	. 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 586	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 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987  ∀wral 3106  ∃wrex 3107  ∃!wreu 3108   ⊆ wss 3881  ∅c0 4243  ∩ cint 4838   class class class wbr 5030  dom cdm 5519  ran crn 5520   “ cima 5522  Oncon0 6159  Fun wfun 6318   Fn wfn 6319  ‘cfv 6324   No csur 33260   <s cslt 33261   bday cbday 33262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-rep 5154  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rmo 3114  df-rab 3115  df-v 3443  df-sbc 3721  df-csb 3829  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-int 4839  df-iun 4883  df-br 5031  df-opab 5093  df-mpt 5111  df-tr 5137  df-id 5425  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-ord 6162  df-on 6163  df-suc 6165  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-1o 8085  df-2o 8086  df-no 33263  df-slt 33264  df-bday 33265

This theorem is referenced by:  conway  33377