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Theorem sucunielr 4236
 Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4256. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc 𝐴𝐵𝐴 𝐵)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2566 . . . 4 (suc 𝐴𝐵 → suc 𝐴 ∈ V)
2 sucexb 4223 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 137 . . 3 (suc 𝐴𝐵𝐴 ∈ V)
4 sucidg 4153 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
53, 4syl 14 . 2 (suc 𝐴𝐵𝐴 ∈ suc 𝐴)
6 elunii 3585 . 2 ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵) → 𝐴 𝐵)
75, 6mpancom 399 1 (suc 𝐴𝐵𝐴 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∈ wcel 1393  Vcvv 2557  ∪ cuni 3580  suc csuc 4102 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-uni 3581  df-suc 4108 This theorem is referenced by: (None)
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