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Theorem eqelsuc 4156
Description: A set belongs to the successor of an equal set. (Contributed by NM, 18-Aug-1994.)
Hypothesis
Ref Expression
eqelsuc.1 𝐴 ∈ V
Assertion
Ref Expression
eqelsuc (𝐴 = 𝐵𝐴 ∈ suc 𝐵)

Proof of Theorem eqelsuc
StepHypRef Expression
1 eqelsuc.1 . . 3 𝐴 ∈ V
21sucid 4154 . 2 𝐴 ∈ suc 𝐴
3 suceq 4139 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
42, 3syl5eleq 2126 1 (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1243  wcel 1393  Vcvv 2557  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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
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-v 2559  df-un 2922  df-sn 3381  df-suc 4108
This theorem is referenced by: (None)
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