Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  oieq2 Structured version   Visualization version   Unicode version

Theorem oieq2 8046
 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 4828 . . . 4
2 seeq2 4812 . . . 4 Se Se
31, 2anbi12d 725 . . 3 Se Se
4 rabeq 3024 . . . . . . 7
54raleqdv 2979 . . . . . . 7
64, 5riotaeqbidv 6273 . . . . . 6
76mpteq2dv 4483 . . . . 5
8 recseq 7110 . . . . 5 recs recs
97, 8syl 17 . . . 4 recs recs
109imaeq1d 5173 . . . . . . 7 recs recs
1110raleqdv 2979 . . . . . 6 recs recs
1211rexeqbi1dv 2982 . . . . 5 recs recs
1312rabbidv 3022 . . . 4 recs recs
149, 13reseq12d 5112 . . 3 recs recs recs recs
153, 14ifbieq1d 3895 . 2 Se recs recs Se recs recs
16 df-oi 8043 . 2 OrdIso Se recs recs
17 df-oi 8043 . 2 OrdIso Se recs recs
1815, 16, 173eqtr4g 2530 1 OrdIso OrdIso
 Colors of variables: wff setvar class Syntax hints:   wn 3   wi 4   wa 376   wceq 1452  wral 2756  wrex 2757  crab 2760  cvv 3031  c0 3722  cif 3872   class class class wbr 4395   cmpt 4454   Se wse 4796   wwe 4797   crn 4840   cres 4841  cima 4842  con0 5430  crio 6269  recscrecs 7107  OrdIsocoi 8042 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-po 4760  df-so 4761  df-fr 4798  df-se 4799  df-we 4800  df-xp 4845  df-cnv 4847  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-pred 5387  df-iota 5553  df-fv 5597  df-riota 6270  df-wrecs 7046  df-recs 7108  df-oi 8043 This theorem is referenced by:  hartogslem1  8075  cantnfval  8191  cantnf0  8198  cantnfres  8200  cantnf  8216  dfac12lem1  8591  dfac12r  8594  hsmexlem2  8875  hsmexlem4  8877  ltbwe  18773
 Copyright terms: Public domain W3C validator