Theorem unvdif 4436

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 4229	. 2 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2		disjdif 4433	. . 3 ⊢ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
3	2	difeq2i 4078	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4		dif0 4334	. 2 ⊢ (V ∖ ∅) = V
5	1, 3, 4	3eqtri 2792	1 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3457   ∖ cdif 3903   ∪ cun 3904   ∩ cin 3905  ∅c0 4286

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 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

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 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287

This theorem is used by:  undif1  4437  dfif4  4505  hashfxnn0  14386  fullfunfnv  36451  hfext  36688