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 u. (_V \ A)) = _V

Proof of Theorem undifv

Step	Hyp	Ref	Expression
1		dfun3 3494	. 2 |- (A u. (_V \ A)) = (_V \ ((_V \ A) i^i (_V \ (_V \ A))))
2		disjdif 3623	. . 3 |- ((_V \ A) i^i (_V \ (_V \ A))) = (/)
3	2	difeq2i 3383	. 2 |- (_V \ ((_V \ A) i^i (_V \ (_V \ A)))) = (_V \ (/))
4		dif0 3621	. 2 |- (_V \ (/)) = _V
5	1, 3, 4	3eqtri 2377	1 |- (A u. (_V \ A)) = _V

Syntax hints:   = wceq 1642  _Vcvv 2860   \ cdif 3207   u. cun 3208   i^i 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