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Theorem sucunielr 4527
Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4548. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc 𝐴𝐵𝐴 𝐵)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2763 . . . 4 (suc 𝐴𝐵 → suc 𝐴 ∈ V)
2 sucexb 4514 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 134 . . 3 (suc 𝐴𝐵𝐴 ∈ V)
4 sucidg 4434 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
53, 4syl 14 . 2 (suc 𝐴𝐵𝐴 ∈ suc 𝐴)
6 elunii 3829 . 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 2160  Vcvv 2752   cuni 3824  suc csuc 4383
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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-rex 2474  df-v 2754  df-un 3148  df-in 3150  df-ss 3157  df-pw 3592  df-sn 3613  df-pr 3614  df-uni 3825  df-suc 4389
This theorem is referenced by:  nnsucuniel  6521
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