Theorem bnj216 30800

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 5804	. 2 ⊢ 𝐵 ∈ suc 𝐵
3		eleq2 2690	. 2 ⊢ (𝐴 = suc 𝐵 → (𝐵 ∈ 𝐴 ↔ 𝐵 ∈ suc 𝐵))
4	2, 3	mpbiri 248	1 ⊢ (𝐴 = suc 𝐵 → 𝐵 ∈ 𝐴)

Syntax hints:   → wi 4   = wceq 1483   ∈ wcel 1990  Vcvv 3200  suc csuc 5725

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-sn 4178  df-suc 5729

This theorem is referenced by:  bnj219  30801  bnj1098  30854  bnj556  30970  bnj557  30971  bnj594  30982  bnj944  31008  bnj966  31014  bnj969  31016  bnj1145  31061