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Theorem oieq1 7481
 Description: Equality theorem for ordinal isomorphism. (Contributed by Mario Carneiro, 23-May-2015.)
Assertion
Ref Expression
oieq1 OrdIso OrdIso

Proof of Theorem oieq1
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 weeq1 4570 . . . 4
2 seeq1 4554 . . . 4 Se Se
31, 2anbi12d 692 . . 3 Se Se
4 breq 4214 . . . . . . . . 9
54ralbidv 2725 . . . . . . . 8
65rabbidv 2948 . . . . . . 7
7 breq 4214 . . . . . . . . 9
87notbid 286 . . . . . . . 8
96, 8raleqbidv 2916 . . . . . . 7
106, 9riotaeqbidv 6552 . . . . . 6
1110mpteq2dv 4296 . . . . 5
12 recseq 6634 . . . . 5 recs recs
1311, 12syl 16 . . . 4 recs recs
1413imaeq1d 5202 . . . . . . 7 recs recs
15 breq 4214 . . . . . . 7
1614, 15raleqbidv 2916 . . . . . 6 recs recs
1716rexbidv 2726 . . . . 5 recs recs
1817rabbidv 2948 . . . 4 recs recs
1913, 18reseq12d 5147 . . 3 recs recs recs recs
20 eqidd 2437 . . 3
213, 19, 20ifbieq12d 3761 . 2 Se recs recs Se recs recs
22 df-oi 7479 . 2 OrdIso Se recs recs
23 df-oi 7479 . 2 OrdIso Se recs recs
2421, 22, 233eqtr4g 2493 1 OrdIso OrdIso
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 359   wceq 1652  wral 2705  wrex 2706  crab 2709  cvv 2956  c0 3628  cif 3739   class class class wbr 4212   cmpt 4266   Se wse 4539   wwe 4540  con0 4581   crn 4879   cres 4880  cima 4881  crio 6542  recscrecs 6632  OrdIsocoi 7478 This theorem is referenced by:  hartogslem1  7511 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-xp 4884  df-cnv 4886  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fv 5462  df-riota 6549  df-recs 6633  df-oi 7479
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