Theorem unvdif 4481

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 4282	. 2 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))))
2		disjdif 4478	. . 3 ⊢ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴))) = ∅
3	2	difeq2i 4133	. 2 ⊢ (V ∖ ((V ∖ 𝐴) ∩ (V ∖ (V ∖ 𝐴)))) = (V ∖ ∅)
4		dif0 4384	. 2 ⊢ (V ∖ ∅) = V
5	1, 3, 4	3eqtri 2767	1 ⊢ (𝐴 ∪ (V ∖ 𝐴)) = V

Syntax hints:   = wceq 1537  Vcvv 3478   ∖ cdif 3960   ∪ cun 3961   ∩ cin 3962  ∅c0 4339

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340

This theorem is referenced by:  undif1  4482  dfif4  4546  hashfxnn0  14373  fullfunfnv  35928  hfext  36165