Theorem compeq 44436

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 3932	. 2 ⊢ (𝑥 ∈ (V ∖ 𝐴) ↔ ¬ 𝑥 ∈ 𝐴)
2	1	eqabi 2864	1 ⊢ (V ∖ 𝐴) = {𝑥 ∣ ¬ 𝑥 ∈ 𝐴}

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2109  {cab 2708  Vcvv 3450   ∖ cdif 3914

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920

This theorem is referenced by: (None)