Theorem clel3g 2977

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 13-Aug-2005.)

Assertion

Ref	Expression
clel3g	⊢ (B ∈ V → (A ∈ B ↔ ∃x(x = B ∧ A ∈ x)))

Distinct variable groups:   x,A   x,B

Allowed substitution hint:   V(x)

Proof of Theorem clel3g

Step	Hyp	Ref	Expression
1		eleq2 2414	. . 3 ⊢ (x = B → (A ∈ x ↔ A ∈ B))
2	1	ceqsexgv 2972	. 2 ⊢ (B ∈ V → (∃x(x = B ∧ A ∈ x) ↔ A ∈ B))
3	2	bicomd 192	1 ⊢ (B ∈ V → (A ∈ B ↔ ∃x(x = B ∧ A ∈ x)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862

This theorem is used by:  clel3  2978  dfiun2g  4000