Theorem velcomp 3977

Description: Characterization of setvar elements of the complement of a class. (Contributed by Andrew Salmon, 15-Jul-2011.)

Assertion

Ref	Expression
velcomp	⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)

Proof of Theorem velcomp

Step	Hyp	Ref	Expression
1		vex 3481	. 2 ⊢ 𝑥 ∈ V
2		eldif 3972	. 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpbiran 709	1 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∈ wcel 2105  Vcvv 3477   ∖ cdif 3959

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-v 3479  df-dif 3965

This theorem is referenced by:  rnep  5939  compeq  44435  compab  44437