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Theorem nocvxmin 31589
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 ∧ ∀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 5441 . . . . . 6 ( bday 𝐴) ⊆ ran bday
2 bdayrn 31575 . . . . . 6 ran bday = On
31, 2sseqtri 3621 . . . . 5 ( bday 𝐴) ⊆ On
4 bdaydm 31576 . . . . . . . . . . 11 dom bday = No
54sseq2i 3614 . . . . . . . . . 10 (𝐴 ⊆ dom bday 𝐴 No )
6 bdayfun 31574 . . . . . . . . . . 11 Fun bday
7 funfvima2 6453 . . . . . . . . . . 11 ((Fun bday 𝐴 ⊆ dom bday ) → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
86, 7mpan 705 . . . . . . . . . 10 (𝐴 ⊆ dom bday → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
95, 8sylbir 225 . . . . . . . . 9 (𝐴 No → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
10 elex2 3205 . . . . . . . . 9 (( bday 𝑥) ∈ ( bday 𝐴) → ∃𝑤 𝑤 ∈ ( bday 𝐴))
119, 10syl6 35 . . . . . . . 8 (𝐴 No → (𝑥𝐴 → ∃𝑤 𝑤 ∈ ( bday 𝐴)))
1211exlimdv 1858 . . . . . . 7 (𝐴 No → (∃𝑥 𝑥𝐴 → ∃𝑤 𝑤 ∈ ( bday 𝐴)))
13 n0 3912 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
14 n0 3912 . . . . . . 7 (( bday 𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( bday 𝐴))
1512, 13, 143imtr4g 285 . . . . . 6 (𝐴 No → (𝐴 ≠ ∅ → ( bday 𝐴) ≠ ∅))
1615impcom 446 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( bday 𝐴) ≠ ∅)
17 onint 6949 . . . . 5 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ ( bday 𝐴))
183, 16, 17sylancr 694 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( bday 𝐴) ∈ ( bday 𝐴))
19 bdayfn 31577 . . . . . 6 bday Fn No
20 fvelimab 6215 . . . . . 6 (( bday Fn No 𝐴 No ) → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2119, 20mpan 705 . . . . 5 (𝐴 No → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2221adantl 482 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2318, 22mpbid 222 . . 3 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))
24233adant3 1079 . 2 ((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴))
25 ssel 3581 . . . . . . . . 9 (𝐴 No → (𝑤𝐴𝑤 No ))
26 ssel 3581 . . . . . . . . 9 (𝐴 No → (𝑡𝐴𝑡 No ))
2725, 26anim12d 585 . . . . . . . 8 (𝐴 No → ((𝑤𝐴𝑡𝐴) → (𝑤 No 𝑡 No )))
2827imp 445 . . . . . . 7 ((𝐴 No ∧ (𝑤𝐴𝑡𝐴)) → (𝑤 No 𝑡 No ))
2928ad2ant2r 782 . . . . . 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 31588 . . . . . . 7 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) → ¬ 𝑤 <s 𝑡))
3130imp 445 . . . . . 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 466 . . . . . . . . 9 ((𝑤𝐴𝑡𝐴) ↔ (𝑡𝐴𝑤𝐴))
33 ancom 466 . . . . . . . . 9 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) ↔ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴)))
3432, 33anbi12i 732 . . . . . . . 8 (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) ↔ ((𝑡𝐴𝑤𝐴) ∧ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴))))
35 nocvxminlem 31588 . . . . . . . 8 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑡𝐴𝑤𝐴) ∧ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴))) → ¬ 𝑡 <s 𝑤))
3634, 35syl5bi 232 . . . . . . 7 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) → ¬ 𝑡 <s 𝑤))
3736imp 445 . . . . . 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 31569 . . . . . . 7 ((𝑤 No 𝑡 No ) → (𝑤 = 𝑡 ↔ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)))
3938biimpar 502 . . . . . 6 (((𝑤 No 𝑡 No ) ∧ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)) → 𝑤 = 𝑡)
4029, 31, 37, 39syl12anc 1321 . . . . 5 (((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) ∧ ((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)))) → 𝑤 = 𝑡)
4140exp32 630 . . . 4 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ((𝑤𝐴𝑡𝐴) → ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡)))
4241ralrimivv 2965 . . 3 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∀𝑤𝐴𝑡𝐴 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡))
43423adant1 1077 . 2 ((𝐴 ≠ ∅ ∧ 𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ∀𝑤𝐴𝑡𝐴 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡))
44 fveq2 6153 . . . 4 (𝑤 = 𝑡 → ( bday 𝑤) = ( bday 𝑡))
4544eqeq1d 2623 . . 3 (𝑤 = 𝑡 → (( bday 𝑤) = ( bday 𝐴) ↔ ( bday 𝑡) = ( bday 𝐴)))
4645reu4 3386 . 2 (∃!𝑤𝐴 ( bday 𝑤) = ( bday 𝐴) ↔ (∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴) ∧ ∀𝑤𝐴𝑡𝐴 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡)))
4724, 43, 46sylanbrc 697 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 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wne 2790  wral 2907  wrex 2908  ∃!wreu 2909  wss 3559  c0 3896   cint 4445   class class class wbr 4618  dom cdm 5079  ran crn 5080  cima 5082  Oncon0 5687  Fun wfun 5846   Fn wfn 5847  cfv 5852   No csur 31529   <s cslt 31530   bday cbday 31531
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4736  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6909
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3191  df-sbc 3422  df-csb 3519  df-dif 3562  df-un 3564  df-in 3566  df-ss 3573  df-pss 3575  df-nul 3897  df-if 4064  df-pw 4137  df-sn 4154  df-pr 4156  df-tp 4158  df-op 4160  df-uni 4408  df-int 4446  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-tr 4718  df-eprel 4990  df-id 4994  df-po 5000  df-so 5001  df-fr 5038  df-we 5040  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-ord 5690  df-on 5691  df-suc 5693  df-iota 5815  df-fun 5854  df-fn 5855  df-f 5856  df-f1 5857  df-fo 5858  df-f1o 5859  df-fv 5860  df-1o 7512  df-2o 7513  df-no 31532  df-slt 31533  df-bday 31534
This theorem is referenced by: (None)
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