Theorem nocvxmin 27838

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 6091	. . . . . 6 ⊢ ( bday “ 𝐴) ⊆ ran bday
2		bdayrn 27835	. . . . . 6 ⊢ ran bday = On
3	1, 2	sseqtri 4032	. . . . 5 ⊢ ( bday “ 𝐴) ⊆ On
4		bdaydm 27834	. . . . . . . . . . 11 ⊢ dom bday = No
5	4	sseq2i 4025	. . . . . . . . . 10 ⊢ (𝐴 ⊆ dom bday ↔ 𝐴 ⊆ No )
6		bdayfun 27832	. . . . . . . . . . 11 ⊢ Fun bday
7		funfvima2 7251	. . . . . . . . . . 11 ⊢ ((Fun bday ∧ 𝐴 ⊆ dom bday ) → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
8	6, 7	mpan 690	. . . . . . . . . 10 ⊢ (𝐴 ⊆ dom bday → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
9	5, 8	sylbir 235	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑥 ∈ 𝐴 → ( bday ‘𝑥) ∈ ( bday “ 𝐴)))
10		elex2 2816	. . . . . . . . 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 1931	. . . . . . 7 ⊢ (𝐴 ⊆ No → (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑤 𝑤 ∈ ( bday “ 𝐴)))
13		n0 4359	. . . . . . 7 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
14		n0 4359	. . . . . . 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 7810	. . . . 5 ⊢ ((( bday “ 𝐴) ⊆ On ∧ ( bday “ 𝐴) ≠ ∅) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
18	3, 16, 17	sylancr 587	. . . 4 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴))
19		bdayfn 27833	. . . . . 6 ⊢ bday Fn No
20		fvelimab 6981	. . . . . 6 ⊢ (( bday Fn No ∧ 𝐴 ⊆ No ) → (∩ ( bday “ 𝐴) ∈ ( bday “ 𝐴) ↔ ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴)))
21	19, 20	mpan 690	. . . . 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 232	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ) → ∃𝑤 ∈ 𝐴 ( bday ‘𝑤) = ∩ ( bday “ 𝐴))
24	23	3adant3 1131	. 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 3989	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → 𝑤 ∈ No ))
26		ssel 3989	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑡 ∈ 𝐴 → 𝑡 ∈ No ))
27	25, 26	anim12d 609	. . . . . . . 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 747	. . . . . 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 27837	. . . . . . 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 628	. . . . . . . 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 27837	. . . . . . . 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 242	. . . . . . 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 27810	. . . . . . 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 837	. . . . 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 3198	. . 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 1129	. 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 6916	. . 3 ⊢ (𝑤 = 𝑡 → (( bday ‘𝑤) = ∩ ( bday “ 𝐴) ↔ ( bday ‘𝑡) = ∩ ( bday “ 𝐴)))
45	44	reu4 3740	. 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 583	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 206   ∧ wa 395   ∧ w3a 1086   = wceq 1537  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∀wral 3059  ∃wrex 3068  ∃!wreu 3376   ⊆ wss 3963  ∅c0 4339  ∩ cint 4951   class class class wbr 5148  dom cdm 5689  ran crn 5690   “ cima 5692  Oncon0 6386  Fun wfun 6557   Fn wfn 6558  ‘cfv 6563   No csur 27699   <s cslt 27700   bday cbday 27701

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-tp 4636  df-op 4638  df-uni 4913  df-int 4952  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1o 8505  df-2o 8506  df-no 27702  df-slt 27703  df-bday 27704

This theorem is referenced by:  conway  27859