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Theorem ordtype2 7503
 Description: For any set-like well-ordered class, if the order isomorphism exists (is a set), then it maps some ordinal onto isomorphically. Otherwise, is a proper class, which implies that either is a proper class or . This weak version of ordtype 7501 does not require the Axiom of Replacement. (Contributed by Mario Carneiro, 25-Jun-2015.)
Hypothesis
Ref Expression
oicl.1 OrdIso
Assertion
Ref Expression
ordtype2 Se

Proof of Theorem ordtype2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2436 . . 3 recs recs
2 eqid 2436 . . 3
3 eqid 2436 . . 3
41, 2, 3ordtypecbv 7486 . 2 recs recs
5 eqid 2436 . 2 recs recs
6 oicl.1 . 2 OrdIso
7 simp1 957 . 2 Se
8 simp2 958 . 2 Se Se
9 simp3 959 . 2 Se
104, 2, 3, 5, 6, 7, 8, 9ordtypelem9 7495 1 Se
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   w3a 936   wceq 1652   wcel 1725  wral 2705  wrex 2706  crab 2709  cvv 2956   class class class wbr 4212   cmpt 4266   cep 4492   Se wse 4539   wwe 4540  con0 4581   cdm 4878   crn 4879  cima 4881   wiso 5455  crio 6542  recscrecs 6632  OrdIsocoi 7478 This theorem is referenced by:  oismo  7509 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-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701 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-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rmo 2713  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-se 4542  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-riota 6549  df-recs 6633  df-oi 7479
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