Theorem velcomp 4054

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 3454	. 2 ⊢ 𝑥 ∈ V
2		eldif 4049	. 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ 𝐴))
3	1, 2	mpbiran 722	1 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∈ wcel 2144  Vcvv 3450   ∖ cdif 4045

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047

This theorem is used by:  ddif  4077  dfun2  4215  dfin2  4216  rnep  5908  compeq  45499  compab  45501