Theorem bnj216 34768

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj216.1	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
bnj216	⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

Proof of Theorem bnj216

Step	Hyp	Ref	Expression
1		bnj216.1	. . 3 ⊢ 𝐵 ∈ V
2	1	sucid 6441	. 2 ⊢ 𝐵 ∈ suc 𝐵
3		eleq2 2824	. 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵))
4	2, 3	mpbiri 258	1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3464  suc csuc 6359

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936  df-sn 4607  df-suc 6363

This theorem is referenced by:  bnj219  34769  bnj1098  34819  bnj556  34936  bnj557  34937  bnj594  34948  bnj944  34974  bnj966  34980  bnj969  34982  bnj1145  35029