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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 2541 . . . 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 3557 . 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 1374  Vcvv 2533  ∪ cuni 3552  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 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 3847  ax-pow 3899  ax-pr 3916  ax-un 4118 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 2288  df-v 2535  df-un 2897  df-in 2899  df-ss 2906  df-pw 3334  df-sn 3354  df-pr 3355  df-uni 3553  df-suc 4055 This theorem is referenced by: (None)
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