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Theorem sucunielr 4256
 Description: Successor and union. The converse (where is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4276. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2611 . . . 4
2 sucexb 4243 . . . 4
31, 2sylibr 132 . . 3
4 sucidg 4173 . . 3
53, 4syl 14 . 2
6 elunii 3608 . 2
75, 6mpancom 413 1
 Colors of variables: wff set class Syntax hints:   wi 4   wcel 1434  cvv 2602  cuni 3603   csuc 4122 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-sep 3898  ax-pow 3950  ax-pr 3966  ax-un 4190 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-rex 2355  df-v 2604  df-un 2978  df-in 2980  df-ss 2987  df-pw 3386  df-sn 3406  df-pr 3407  df-uni 3604  df-suc 4128 This theorem is referenced by:  nnsucuniel  6132
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