Theorem eqelsuc 4420

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 4418	. 2 ⊢ 𝐴 ∈ suc 𝐴
3		suceq 4403	. 2 ⊢ (𝐴 = 𝐵 → suc 𝐴 = suc 𝐵)
4	2, 3	eleqtrid 2266	1 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ suc 𝐵)

Syntax hints:   → wi 4   = wceq 1353   ∈ wcel 2148  Vcvv 2738  suc csuc 4366

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-un 3134  df-sn 3599  df-suc 4372

This theorem is referenced by: (None)