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Theorem sucunielr 4183
Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4198. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc A BA B)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2542 . . . 4 (suc A B → suc A V)
2 sucexb 4171 . . . 4 (A V ↔ suc A V)
31, 2sylibr 137 . . 3 (suc A BA V)
4 sucidg 4100 . . 3 (A V → A suc A)
53, 4syl 14 . 2 (suc A BA suc A)
6 elunii 3558 . 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 2534   cuni 3553  suc csuc 4049
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 1316  ax-7 1317  ax-gen 1318  ax-ie1 1364  ax-ie2 1365  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 2005  ax-sep 3848  ax-pow 3900  ax-pr 3917  ax-un 4118
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1629  df-clab 2010  df-cleq 2016  df-clel 2019  df-nfc 2150  df-rex 2289  df-v 2536  df-un 2898  df-in 2900  df-ss 2907  df-pw 3335  df-sn 3355  df-pr 3356  df-uni 3554  df-suc 4055
This theorem is referenced by: (None)
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