Theorem eqelsuc 6332

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 = 𝐵 → 𝐴 ∈ suc 𝐵)

Proof of Theorem eqelsuc

Step	Hyp	Ref	Expression
1		eqelsuc.1	. . 3 ⊢ 𝐴 ∈ V
2	1	sucid 6330	. 2 ⊢ 𝐴 ∈ suc 𝐴
3		suceq 6316	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4	2, 3	eleqtrid 2845	1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   = wceq 1539   ∈ wcel 2108  Vcvv 3422  suc csuc 6253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-un 3888  df-sn 4559  df-suc 6257

This theorem is referenced by:  pssnn  8913  pssnnOLD  8969