Theorem unvdif 4429

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 4222	. 2 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2		disjdif 4426	. . 3 ⊢ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
3	2	difeq2i 4071	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4		dif0 4327	. 2 ⊢ (V ∖ ∅) = V
5	1, 3, 4	3eqtri 2787	1 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V

This proof depends on syntax axioms:   = wceq 1570  Vcvv 3450   ∖ cdif 3896   ∪ cun 3897   ∩ cin 3898  ∅c0 4279

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 2147  ax-9 2155  ax-ext 2732

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 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280

This theorem is used by:  undif1  4430  dfif4  4498  hashfxnn0  14401  fullfunfnv  36525  hfext  36763