Theorem unvdif 4432

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 4224	. 2 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2		disjdif 4429	. . 3 ⊢ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
3	2	difeq2i 4063	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4		dif0 4330	. 2 ⊢ (V ∖ ∅) = V
5	1, 3, 4	3eqtri 2790	1 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3454   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282

This theorem is used by:  undif1  4433  dfif4  4501  hashfxnn0  14522  fullfunfnv  36925  hfext  37149