Theorem undifv 3625

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
undifv	⊢ (A ∪ (V ∖ A)) = V

Proof of Theorem undifv

Step	Hyp	Ref	Expression
1		dfun3 3494	. 2 ⊢ (A ∪ (V ∖ A)) = (V ∖ ((V ∖ A) ∩ (V ∖ (V ∖ A))))
2		disjdif 3623	. . 3 ⊢ ((V ∖ A) ∩ (V ∖ (V ∖ A))) = ∅
3	2	difeq2i 3383	. 2 ⊢ (V ∖ ((V ∖ A) ∩ (V ∖ (V ∖ A)))) = (V ∖ ∅)
4		dif0 3621	. 2 ⊢ (V ∖ ∅) = V
5	1, 3, 4	3eqtri 2377	1 ⊢ (A ∪ (V ∖ A)) = V

Syntax hints:   = wceq 1642  Vcvv 2860   ∖ cdif 3207   ∪ cun 3208   ∩ cin 3209  ∅c0 3551

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  undif1  3626  dfif4  3674