MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqelsuc Structured version   Visualization version   GIF version

Theorem eqelsuc 6468
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 6466 . 2 𝐴 ∈ suc 𝐴
3 suceq 6450 . 2 (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
42, 3eleqtrid 2847 1 (𝐴 = 𝐵𝐴 ∈ suc 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  wcel 2108  Vcvv 3480  suc csuc 6386
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-sn 4627  df-suc 6390
This theorem is referenced by:  pssnn  9208
  Copyright terms: Public domain W3C validator