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Theorem sucunielr 4201
 Description: Successor and union. The converse (where B is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4216. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc A BA B)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2560 . . . 4 (suc A B → suc A V)
2 sucexb 4189 . . . 4 (A V ↔ suc A V)
31, 2sylibr 137 . . 3 (suc A BA V)
4 sucidg 4119 . . 3 (A V → A suc A)
53, 4syl 14 . 2 (suc A BA suc A)
6 elunii 3576 . 2 ((A suc A suc A B) → A B)
75, 6mpancom 399 1 (suc A BA B)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1390  Vcvv 2551  ∪ cuni 3571  suc csuc 4068 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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-uni 3572  df-suc 4074 This theorem is referenced by: (None)
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