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Theorem sucunielr 4386
Description: Successor and union. The converse (where 𝐵 is an ordinal) implies excluded middle, as seen at ordsucunielexmid 4406. (Contributed by Jim Kingdon, 2-Aug-2019.)
Assertion
Ref Expression
sucunielr (suc 𝐴𝐵𝐴 𝐵)

Proof of Theorem sucunielr
StepHypRef Expression
1 elex 2668 . . . 4 (suc 𝐴𝐵 → suc 𝐴 ∈ V)
2 sucexb 4373 . . . 4 (𝐴 ∈ V ↔ suc 𝐴 ∈ V)
31, 2sylibr 133 . . 3 (suc 𝐴𝐵𝐴 ∈ V)
4 sucidg 4298 . . 3 (𝐴 ∈ V → 𝐴 ∈ suc 𝐴)
53, 4syl 14 . 2 (suc 𝐴𝐵𝐴 ∈ suc 𝐴)
6 elunii 3707 . 2 ((𝐴 ∈ suc 𝐴 ∧ suc 𝐴𝐵) → 𝐴 𝐵)
75, 6mpancom 416 1 (suc 𝐴𝐵𝐴 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wcel 1463  Vcvv 2657   cuni 3702  suc csuc 4247
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-rex 2396  df-v 2659  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-uni 3703  df-suc 4253
This theorem is referenced by:  nnsucuniel  6345
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