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Theorem sucunielr 4505
Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4526. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc 𝐴𝐵𝐴 𝐵)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2748 . . . 4 (suc 𝐴𝐵 → suc 𝐴 ∈ V)
2 sucexb 4492 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 134 . . 3 (suc 𝐴𝐵𝐴 ∈ V)
4 sucidg 4412 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
53, 4syl 14 . 2 (suc 𝐴𝐵𝐴 ∈ suc 𝐴)
6 elunii 3812 . 2 ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵) → 𝐴 𝐵)
75, 6mpancom 422 1 (suc 𝐴𝐵𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 2148  Vcvv 2737   cuni 3807  suc csuc 4361
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-pow 4171  ax-pr 4205  ax-un 4429
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-rex 2461  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pw 3576  df-sn 3597  df-pr 3598  df-uni 3808  df-suc 4367
This theorem is referenced by:  nnsucuniel  6489
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