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Theorem oieq2 7485
 Description: Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
Assertion
Ref Expression
oieq2 OrdIso OrdIso

Proof of Theorem oieq2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 weeq2 4574 . . . 4
2 seeq2 4558 . . . 4 Se Se
31, 2anbi12d 693 . . 3 Se Se
4 rabeq 2952 . . . . . . 7
54raleqdv 2912 . . . . . . 7
64, 5riotaeqbidv 6555 . . . . . 6
76mpteq2dv 4299 . . . . 5
8 recseq 6637 . . . . 5 recs recs
97, 8syl 16 . . . 4 recs recs
109imaeq1d 5205 . . . . . . 7 recs recs
1110raleqdv 2912 . . . . . 6 recs recs
1211rexeqbi1dv 2915 . . . . 5 recs recs
1312rabbidv 2950 . . . 4 recs recs
149, 13reseq12d 5150 . . 3 recs recs recs recs
15 eqidd 2439 . . 3
163, 14, 15ifbieq12d 3763 . 2 Se recs recs Se recs recs
17 df-oi 7482 . 2 OrdIso Se recs recs
18 df-oi 7482 . 2 OrdIso Se recs recs
1916, 17, 183eqtr4g 2495 1 OrdIso OrdIso
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 360   wceq 1653  wral 2707  wrex 2708  crab 2711  cvv 2958  c0 3630  cif 3741   class class class wbr 4215   cmpt 4269   Se wse 4542   wwe 4543  con0 4584   crn 4882   cres 4883  cima 4884  crio 6545  recscrecs 6635  OrdIsocoi 7481 This theorem is referenced by:  hartogslem1  7514  cantnfval  7626  cantnf0  7633  cantnfres  7636  cantnf  7652  dfac12lem1  8028  dfac12r  8031  hsmexlem2  8312  hsmexlem4  8314  ltbwe  16538 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-mpt 4271  df-po 4506  df-so 4507  df-fr 4544  df-se 4545  df-we 4546  df-xp 4887  df-cnv 4889  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fv 5465  df-riota 6552  df-recs 6636  df-oi 7482
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