Theorem clel3 3592

Description: Alternate definition of membership in 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 3591	. 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 205   ∧ wa 396   = wceq 1539  ∃wex 1782   ∈ wcel 2106  Vcvv 3432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816

This theorem is referenced by:  uniprOLD  4858  elold  34053  brcup  34241  brcap  34242