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Theorem sucunielr 4290
Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4310. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc 𝐴𝐵𝐴 𝐵)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2621 . . . 4 (suc 𝐴𝐵 → suc 𝐴 ∈ V)
2 sucexb 4277 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 132 . . 3 (suc 𝐴𝐵𝐴 ∈ V)
4 sucidg 4207 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
53, 4syl 14 . 2 (suc 𝐴𝐵𝐴 ∈ suc 𝐴)
6 elunii 3632 . 2 ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵) → 𝐴 𝐵)
75, 6mpancom 413 1 (suc 𝐴𝐵𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1434  Vcvv 2612   cuni 3627  suc csuc 4156
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 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-rex 2359  df-v 2614  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-uni 3628  df-suc 4162
This theorem is referenced by:  nnsucuniel  6188
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