Theorem compeq 41947

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 3898	. 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
2	1	abbi2i 2878	1 ⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1539   ∈ wcel 2108  {cab 2715  Vcvv 3422   ∖ cdif 3880

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886

This theorem is referenced by: (None)