Theorem compeq 45562

Description: Equality between two ways of saying "the complement of 𝐴". (Contributed by Andrew Salmon, 15-Jul-2011.)

Assertion

Ref	Expression
compeq	⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴}

Distinct variable group:   𝑥,𝐴

Proof of Theorem compeq

Step	Hyp	Ref	Expression
1		velcomp 4056	. 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
2	1	eqabi 2898	1 ⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴}

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1569   ∈ wcel 2146  {cab 2741  Vcvv 3453   ∖ cdif 4047

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 1944  ax-6 2001  ax-7 2042  ax-8 2148  ax-9 2156  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2101  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3455  df-dif 4049

This theorem is used by: (None)