Theorem unvdif 4434

Description: The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
unvdif	⊢ (𝐴 ∪ (V ∖ 𝐴)) = V

Proof of Theorem unvdif

Step	Hyp	Ref	Expression
1		dfun3 4225	. 2 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2		disjdif 4431	. . 3 ⊢ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
3	2	difeq2i 4079	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4		dif0 4332	. 2 ⊢ (V ∖ ∅) = V
5	1, 3, 4	3eqtri 2768	1 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V

Syntax hints:   = wceq 1541  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909  ∅c0 4282

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283

This theorem is referenced by:  undif1  4435  dfif4  4501  hashfxnn0  14237  fullfunfnv  34531  hfext  34768