Theorem eqelsuc 6266

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 6264	. 2 ⊢ 𝐴 ∈ suc 𝐴
3		suceq 6250	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4	2, 3	eleqtrid 2919	1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   = wceq 1533   ∈ wcel 2110  Vcvv 3494  suc csuc 6187

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 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-un 3940  df-sn 4561  df-suc 6191

This theorem is referenced by:  pssnn  8730