Theorem clel3 2978

Description: An alternate definition of class membership when the class is a set. (Contributed by NM, 18-Aug-1993.)

Hypothesis

Ref	Expression
clel3.1	⊢ B ∈ V

Assertion

Ref	Expression
clel3	⊢ (A ∈ B ↔ ∃x(x = B ∧ A ∈ x))

Distinct variable groups:   x,A   x,B

Proof of Theorem clel3

Step	Hyp	Ref	Expression
1		clel3.1	. 2 ⊢ B ∈ V
2		clel3g 2977	. 2 ⊢ (B ∈ V → (A ∈ B ↔ ∃x(x = B ∧ A ∈ x)))
3	1, 2	ax-mp 5	1 ⊢ (A ∈ B ↔ ∃x(x = B ∧ A ∈ x))

Syntax hints:   ↔ wb 176   ∧ wa 358  ∃wex 1541   = wceq 1642   ∈ wcel 1710  Vcvv 2860

This theorem was proved from 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 theorem 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 referenced by:  unipr  3906  nnsucelrlem1  4425  tfinnnlem1  4534  dfop2lem1  4574  setconslem2  4733  nenpw1pwlem1  6085