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Theorem nocvxmin 27729
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 ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃!𝑤𝐴 ( 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.
StepHypRef Expression
1 imassrn 6077 . . . . . 6 ( bday 𝐴) ⊆ ran bday
2 bdayrn 27726 . . . . . 6 ran bday = On
31, 2sseqtri 4016 . . . . 5 ( bday 𝐴) ⊆ On
4 bdaydm 27725 . . . . . . . . . . 11 dom bday = No
54sseq2i 4009 . . . . . . . . . 10 (𝐴 ⊆ dom bday 𝐴 No )
6 bdayfun 27723 . . . . . . . . . . 11 Fun bday
7 funfvima2 7247 . . . . . . . . . . 11 ((Fun bday 𝐴 ⊆ dom bday ) → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
86, 7mpan 688 . . . . . . . . . 10 (𝐴 ⊆ dom bday → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
95, 8sylbir 234 . . . . . . . . 9 (𝐴 No → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
10 elex2 2807 . . . . . . . . 9 (( bday 𝑥) ∈ ( bday 𝐴) → ∃𝑤 𝑤 ∈ ( bday 𝐴))
119, 10syl6 35 . . . . . . . 8 (𝐴 No → (𝑥𝐴 → ∃𝑤 𝑤 ∈ ( bday 𝐴)))
1211exlimdv 1928 . . . . . . 7 (𝐴 No → (∃𝑥 𝑥𝐴 → ∃𝑤 𝑤 ∈ ( bday 𝐴)))
13 n0 4348 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
14 n0 4348 . . . . . . 7 (( bday 𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( bday 𝐴))
1512, 13, 143imtr4g 295 . . . . . 6 (𝐴 No → (𝐴 ≠ ∅ → ( bday 𝐴) ≠ ∅))
1615impcom 406 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( bday 𝐴) ≠ ∅)
17 onint 7797 . . . . 5 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ ( bday 𝐴))
183, 16, 17sylancr 585 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( bday 𝐴) ∈ ( bday 𝐴))
19 bdayfn 27724 . . . . . 6 bday Fn No
20 fvelimab 6974 . . . . . 6 (( bday Fn No 𝐴 No ) → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2119, 20mpan 688 . . . . 5 (𝐴 No → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2221adantl 480 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2318, 22mpbid 231 . . 3 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))
24233adant3 1129 . 2 ((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))
25 ssel 3973 . . . . . . . . 9 (𝐴 No → (𝑤𝐴𝑤 No ))
26 ssel 3973 . . . . . . . . 9 (𝐴 No → (𝑡𝐴𝑡 No ))
2725, 26anim12d 607 . . . . . . . 8 (𝐴 No → ((𝑤𝐴𝑡𝐴) → (𝑤 No 𝑡 No )))
2827imp 405 . . . . . . 7 ((𝐴 No ∧ (𝑤𝐴𝑡𝐴)) → (𝑤 No 𝑡 No ))
2928ad2ant2r 745 . . . . . 6 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)))) → (𝑤 No 𝑡 No ))
30 nocvxminlem 27728 . . . . . . 7 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) → ¬ 𝑤 <s 𝑡))
3130imp 405 . . . . . 6 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)))) → ¬ 𝑤 <s 𝑡)
32 ancom 459 . . . . . . . . 9 ((𝑤𝐴𝑡𝐴) ↔ (𝑡𝐴𝑤𝐴))
33 ancom 459 . . . . . . . . 9 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) ↔ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴)))
3432, 33anbi12i 626 . . . . . . . 8 (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) ↔ ((𝑡𝐴𝑤𝐴) ∧ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴))))
35 nocvxminlem 27728 . . . . . . . 8 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑡𝐴𝑤𝐴) ∧ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴))) → ¬ 𝑡 <s 𝑤))
3634, 35biimtrid 241 . . . . . . 7 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) → ¬ 𝑡 <s 𝑤))
3736imp 405 . . . . . 6 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)))) → ¬ 𝑡 <s 𝑤)
38 slttrieq2 27701 . . . . . . 7 ((𝑤 No 𝑡 No ) → (𝑤 = 𝑡 ↔ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)))
3938biimpar 476 . . . . . 6 (((𝑤 No 𝑡 No ) ∧ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)) → 𝑤 = 𝑡)
4029, 31, 37, 39syl12anc 835 . . . . 5 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)))) → 𝑤 = 𝑡)
4140exp32 419 . . . 4 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ((𝑤𝐴𝑡𝐴) → ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡)))
4241ralrimivv 3194 . . 3 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∀𝑤𝐴𝑡𝐴 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡))
43423adant1 1127 . 2 ((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∀𝑤𝐴𝑡𝐴 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡))
44 fveqeq2 6909 . . 3 (𝑤 = 𝑡 → (( bday 𝑤) = ( bday 𝐴) ↔ ( bday 𝑡) = ( bday 𝐴)))
4544reu4 3726 . 2 (∃!𝑤𝐴 ( bday 𝑤) = ( bday 𝐴) ↔ (∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴) ∧ ∀𝑤𝐴𝑡𝐴 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡)))
4624, 43, 45sylanbrc 581 1 ((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃!𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2936  wral 3057  wrex 3066  ∃!wreu 3370  wss 3947  c0 4324   cint 4951   class class class wbr 5150  dom cdm 5680  ran crn 5681  cima 5683  Oncon0 6372  Fun wfun 6545   Fn wfn 6546  cfv 6551   No csur 27591   <s cslt 27592   bday cbday 27593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2698  ax-rep 5287  ax-sep 5301  ax-nul 5308  ax-pr 5431  ax-un 7744
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2937  df-ral 3058  df-rex 3067  df-rmo 3372  df-reu 3373  df-rab 3429  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 4325  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-tp 4635  df-op 4637  df-uni 4911  df-int 4952  df-iun 5000  df-br 5151  df-opab 5213  df-mpt 5234  df-tr 5268  df-id 5578  df-eprel 5584  df-po 5592  df-so 5593  df-fr 5635  df-we 5637  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-ord 6375  df-on 6376  df-suc 6378  df-iota 6503  df-fun 6553  df-fn 6554  df-f 6555  df-f1 6556  df-fo 6557  df-f1o 6558  df-fv 6559  df-1o 8491  df-2o 8492  df-no 27594  df-slt 27595  df-bday 27596
This theorem is referenced by:  conway  27750
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