Theorem eqelsuc 4174

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 4172	. 2 ⊢ 𝐴 ∈ suc 𝐴
3		suceq 4157	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4	2, 3	syl5eleq 2167	1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   = wceq 1284   ∈ wcel 1433  Vcvv 2601  suc csuc 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-un 2977  df-sn 3404  df-suc 4126

This theorem is referenced by: (None)