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Theorem sucunielr 4185
Description: Successor and union. The converse (where is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4200. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr  suc  U.

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2543 . . . 4  suc  suc  _V
2 sucexb 4173 . . . 4  _V  suc  _V
31, 2sylibr 137 . . 3  suc  _V
4 sucidg 4102 . . 3  _V  suc
53, 4syl 14 . 2  suc  suc
6 elunii 3559 . 2  suc  suc  U.
75, 6mpancom 401 1  suc  U.
Colors of variables: wff set class
Syntax hints:   wi 4   wcel 1374   _Vcvv 2535   U.cuni 3554   suc csuc 4051
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 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120
This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-rex 2290  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-uni 3555  df-suc 4057
This theorem is referenced by: (None)
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