Theorem nocvxmin 28198

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		nobdaymin 28196	. . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ≠ ∅) → ∃𝑤 ∈ 𝐴 (bday‘𝑤) = ∩ (bday “ 𝐴))
2	1	ancoms 464	. . 3 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No) → ∃𝑤 ∈ 𝐴 (bday‘𝑤) = ∩ (bday “ 𝐴))
3	2	3adant3 1150	. 2 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃𝑤 ∈ 𝐴 (bday‘𝑤) = ∩ (bday “ 𝐴))
4		ssel 3907	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑤 ∈ 𝐴 → 𝑤 ∈ No))
5		ssel 3907	. . . . . . . . 9 ⊢ (𝐴 ⊆ No → (𝑡 ∈ 𝐴 → 𝑡 ∈ No))
6	4, 5	anim12d 621	. . . . . . . 8 ⊢ (𝐴 ⊆ No → ((𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴) → (𝑤 ∈ No ∧ 𝑡 ∈ No)))
7	6	imp 412	. . . . . . 7 ⊢ ((𝐴 ⊆ No ∧ (𝑤 ∈ 𝐴 ∧ 𝑡 ∈ 𝐴)) → (𝑤 ∈ No ∧ 𝑡 ∈ No))
8	7	ad2ant2r 760	. . . . . 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))
9		nocvxminlem 28197	. . . . . . 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))
10	9	imp 412	. . . . . 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)
11		an2anr 648	. . . . . . . 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))))
12		nocvxminlem 28197	. . . . . . . 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))
13	11, 12	biimtrid 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))
14	13	imp 412	. . . . . 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)
15		ltstrieq2 28164	. . . . . . 7 ⊢ ((𝑤 ∈ No ∧ 𝑡 ∈ No) → (𝑤 = 𝑡 ↔ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)))
16	15	biimpar 483	. . . . . 6 ⊢ (((𝑤 ∈ No ∧ 𝑡 ∈ No) ∧ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)) → 𝑤 = 𝑡)
17	8, 10, 14, 16	syl12anc 850	. . . . 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)
18	17	exp32 426	. . . 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)))
19	18	ralrimivv 3207	. . 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))
20	19	3adant1 1148	. 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))
21		fveqeq2 6925	. . 3 ⊢ (𝑤 = 𝑡 → ((bday‘𝑤) = ∩ (bday “ 𝐴) ↔ (bday‘𝑡) = ∩ (bday “ 𝐴)))
22	21	reu4 3692	. 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)))
23	3, 20, 22	sylanbrc 595	1 ⊢ ((𝐴 ≠ ∅ ∧ 𝐴 ⊆ No ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ No ((𝑥 <s 𝑧 ∧ 𝑧 <s 𝑦) → 𝑧 ∈ 𝐴)) → ∃!𝑤 ∈ 𝐴 (bday‘𝑤) = ∩ (bday “ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  ∃!wreu 3367   ⊆ wss 3900  ∅c0 4281  ∩ cint 4912   class class class wbr 5108   “ cima 5661  ‘cfv 6566  Nocsur 28054   <s clts 28055  bdaycbday 28056

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pow 5334  ax-pr 5398  ax-un 7783

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3456  df-sbc 3743  df-csb 3852  df-ss 3901  df-pss 4017  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-tp 4593  df-op 4841  df-uni 4873  df-int 4913  df-br 5109  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5553  df-eprel 5558  df-po 5566  df-so 5567  df-fr 5611  df-we 5613  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-ord 6391  df-on 6392  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-fo 6572  df-fv 6574  df-1o 8514  df-2o 8515  df-no 28057  df-lts 28058  df-bday 28059

This theorem is used by:  conway  28222