Theorem bnj216 35030

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 6432	. 2 ⊢ 𝐵 ∈ suc 𝐵
3		eleq2 2853	. 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵))
4	2, 3	mpbiri 260	1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1562   ∈ wcel 2144  Vcvv 3456  suc csuc 6350

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-ext 2736

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1565  df-ex 1802  df-sb 2093  df-clab 2743  df-cleq 2756  df-clel 2839  df-v 3458  df-un 3911  df-sn 4585  df-suc 6354

This theorem is referenced by:  bnj219  35031  bnj1098  35081  bnj556  35197  bnj557  35198  bnj594  35209  bnj944  35235  bnj966  35241  bnj969  35243  bnj1145  35290