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Theorem eqelsuc 6470
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 6468 . 2 𝐴 ∈ suc 𝐴
3 suceq 6452 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
42, 3eleqtrid 2845 1 (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  Vcvv 3478  suc csuc 6388
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-un 3968  df-sn 4632  df-suc 6392
This theorem is referenced by:  pssnn  9207
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