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Theorem eqelsuc 6455
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 6453 . 2 𝐴 ∈ suc 𝐴
3 suceq 6437 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
42, 3eleqtrid 2831 1 (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1533  wcel 2098  Vcvv 3461  suc csuc 6373
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-v 3463  df-un 3949  df-sn 4631  df-suc 6377
This theorem is referenced by:  pssnn  9193  pssnnOLD  9290
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