Theorem clel3 3655

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 3654	. 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 208   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-ex 1781  df-cleq 2814  df-clel 2893

This theorem is referenced by:  unipr  4855  brcup  33400  brcap  33401