Theorem bnj216 35283

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 6437	. 2 ⊢ 𝐵 ∈ suc 𝐵
3		eleq2 2849	. 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵))
4	2, 3	mpbiri 261	1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450  suc csuc 6354

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-un 3904  df-sn 4585  df-suc 6358

This theorem is used by:  bnj219  35284  bnj1098  35334  bnj556  35450  bnj557  35451  bnj594  35462  bnj944  35488  bnj966  35494  bnj969  35496  bnj1145  35543