Theorem eqelsuc 4484

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 4482	. 2 ⊢ 𝐴 ∈ suc 𝐴
3		suceq 4467	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4	2, 3	eleqtrid 2296	1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   = wceq 1373   ∈ wcel 2178  Vcvv 2776  suc csuc 4430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-suc 4436

This theorem is referenced by: (None)