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Theorem sucunielr 4158
Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4173. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc A BA B)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2541 . . . 4 (suc A B → suc A V)
2 sucexb 4146 . . . 4 (A V ↔ suc A V)
31, 2sylibr 137 . . 3 (suc A BA V)
4 sucidg 4075 . . 3 (A V → A suc A)
53, 4syl 14 . 2 (suc A BA suc A)
6 elunii 3537 . 2 ((A suc A suc A B) → A B)
75, 6mpancom 401 1 (suc A BA B)
Colors of variables: wff set class
Syntax hints:  wi 4   wcel 1375  Vcvv 2533   cuni 3532  suc csuc 4026
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1315  ax-7 1316  ax-gen 1317  ax-ie1 1362  ax-ie2 1363  ax-8 1377  ax-10 1378  ax-11 1379  ax-i12 1380  ax-bnd 1381  ax-4 1382  ax-13 1386  ax-14 1387  ax-17 1401  ax-i9 1405  ax-ial 1410  ax-i5r 1411  ax-ext 2004  ax-sep 3827  ax-pow 3879  ax-pr 3896  ax-un 4093
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1329  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2288  df-v 2535  df-un 2900  df-in 2902  df-ss 2909  df-pw 3313  df-sn 3333  df-pr 3334  df-uni 3533  df-suc 4031
This theorem is referenced by: (None)
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