Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  nocvxmin Structured version   Visualization version   GIF version

Theorem nocvxmin 33256
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 5913 . . . . . 6 ( bday 𝐴) ⊆ ran bday
2 bdayrn 33253 . . . . . 6 ran bday = On
31, 2sseqtri 3979 . . . . 5 ( bday 𝐴) ⊆ On
4 bdaydm 33252 . . . . . . . . . . 11 dom bday = No
54sseq2i 3972 . . . . . . . . . 10 (𝐴 ⊆ dom bday 𝐴 No )
6 bdayfun 33250 . . . . . . . . . . 11 Fun bday
7 funfvima2 6967 . . . . . . . . . . 11 ((Fun bday 𝐴 ⊆ dom bday ) → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
86, 7mpan 689 . . . . . . . . . 10 (𝐴 ⊆ dom bday → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
95, 8sylbir 238 . . . . . . . . 9 (𝐴 No → (𝑥𝐴 → ( bday 𝑥) ∈ ( bday 𝐴)))
10 elex2 3493 . . . . . . . . 9 (( bday 𝑥) ∈ ( bday 𝐴) → ∃𝑤 𝑤 ∈ ( bday 𝐴))
119, 10syl6 35 . . . . . . . 8 (𝐴 No → (𝑥𝐴 → ∃𝑤 𝑤 ∈ ( bday 𝐴)))
1211exlimdv 1935 . . . . . . 7 (𝐴 No → (∃𝑥 𝑥𝐴 → ∃𝑤 𝑤 ∈ ( bday 𝐴)))
13 n0 4283 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
14 n0 4283 . . . . . . 7 (( bday 𝐴) ≠ ∅ ↔ ∃𝑤 𝑤 ∈ ( bday 𝐴))
1512, 13, 143imtr4g 299 . . . . . 6 (𝐴 No → (𝐴 ≠ ∅ → ( bday 𝐴) ≠ ∅))
1615impcom 411 . . . . 5 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( bday 𝐴) ≠ ∅)
17 onint 7485 . . . . 5 ((( bday 𝐴) ⊆ On ∧ ( bday 𝐴) ≠ ∅) → ( bday 𝐴) ∈ ( bday 𝐴))
183, 16, 17sylancr 590 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( bday 𝐴) ∈ ( bday 𝐴))
19 bdayfn 33251 . . . . . 6 bday Fn No
20 fvelimab 6710 . . . . . 6 (( bday Fn No 𝐴 No ) → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2119, 20mpan 689 . . . . 5 (𝐴 No → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2221adantl 485 . . . 4 ((𝐴 ≠ ∅ ∧ 𝐴 No ) → ( ( bday 𝐴) ∈ ( bday 𝐴) ↔ ∃𝑤𝐴 ( bday 𝑤) = ( bday 𝐴)))
2318, 22mpbid 235 . . 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 3937 . . . . . . . . 9 (𝐴 No → (𝑤𝐴𝑤 No ))
26 ssel 3937 . . . . . . . . 9 (𝐴 No → (𝑡𝐴𝑡 No ))
2725, 26anim12d 611 . . . . . . . 8 (𝐴 No → ((𝑤𝐴𝑡𝐴) → (𝑤 No 𝑡 No )))
2827imp 410 . . . . . . 7 ((𝐴 No ∧ (𝑤𝐴𝑡𝐴)) → (𝑤 No 𝑡 No ))
2928ad2ant2r 746 . . . . . 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 33255 . . . . . . 7 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) → ¬ 𝑤 <s 𝑡))
3130imp 410 . . . . . 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 464 . . . . . . . . 9 ((𝑤𝐴𝑡𝐴) ↔ (𝑡𝐴𝑤𝐴))
33 ancom 464 . . . . . . . . 9 ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) ↔ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴)))
3432, 33anbi12i 629 . . . . . . . 8 (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) ↔ ((𝑡𝐴𝑤𝐴) ∧ (( bday 𝑡) = ( bday 𝐴) ∧ ( bday 𝑤) = ( bday 𝐴))))
35 nocvxminlem 33255 . . . . . . . 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 245 . . . . . . 7 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → (((𝑤𝐴𝑡𝐴) ∧ (( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴))) → ¬ 𝑡 <s 𝑤))
3736imp 410 . . . . . 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 33237 . . . . . . 7 ((𝑤 No 𝑡 No ) → (𝑤 = 𝑡 ↔ (¬ 𝑤 <s 𝑡 ∧ ¬ 𝑡 <s 𝑤)))
3938biimpar 481 . . . . . 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 424 . . . 4 ((𝐴 No ∧ ∀𝑥𝐴𝑦𝐴𝑧 No ((𝑥 <s 𝑧𝑧 <s 𝑦) → 𝑧𝐴)) → ((𝑤𝐴𝑡𝐴) → ((( bday 𝑤) = ( bday 𝐴) ∧ ( bday 𝑡) = ( bday 𝐴)) → 𝑤 = 𝑡)))
4241ralrimivv 3178 . . 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 6652 . . 3 (𝑤 = 𝑡 → (( bday 𝑤) = ( bday 𝐴) ↔ ( bday 𝑡) = ( bday 𝐴)))
4544reu4 3699 . 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 586 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 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wne 3007  wral 3126  wrex 3127  ∃!wreu 3128  wss 3910  c0 4266   cint 4849   class class class wbr 5039  dom cdm 5528  ran crn 5529  cima 5531  Oncon0 6164  Fun wfun 6322   Fn wfn 6323  cfv 6328   No csur 33155   <s cslt 33156   bday cbday 33157
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-rep 5163  ax-sep 5176  ax-nul 5183  ax-pow 5239  ax-pr 5303  ax-un 7436
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 2071  df-mo 2623  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-ne 3008  df-ral 3131  df-rex 3132  df-reu 3133  df-rmo 3134  df-rab 3135  df-v 3473  df-sbc 3750  df-csb 3858  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4267  df-if 4441  df-pw 4514  df-sn 4541  df-pr 4543  df-tp 4545  df-op 4547  df-uni 4812  df-int 4850  df-iun 4894  df-br 5040  df-opab 5102  df-mpt 5120  df-tr 5146  df-id 5433  df-eprel 5438  df-po 5447  df-so 5448  df-fr 5487  df-we 5489  df-xp 5534  df-rel 5535  df-cnv 5536  df-co 5537  df-dm 5538  df-rn 5539  df-res 5540  df-ima 5541  df-ord 6167  df-on 6168  df-suc 6170  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-1o 8077  df-2o 8078  df-no 33158  df-slt 33159  df-bday 33160
This theorem is referenced by:  conway  33272
  Copyright terms: Public domain W3C validator