Theorem unvdif 4042

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 u. ( _V \ A ) ) = _V

Proof of Theorem unvdif

Step	Hyp	Ref	Expression
1		dfun3 3865	. 2 |- ( A u. ( _V \ A ) ) = ( _V \ ( ( _V \ A ) i^i ( _V \ ( _V \ A ) ) ) )
2		disjdif 4040	. . 3 |- ( ( _V \ A ) i^i ( _V \ ( _V \ A ) ) ) = (/)
3	2	difeq2i 3725	. 2 |- ( _V \ ( ( _V \ A ) i^i ( _V \ ( _V \ A ) ) ) ) = ( _V \ (/) )
4		dif0 3950	. 2 |- ( _V \ (/) ) = _V
5	1, 3, 4	3eqtri 2648	1 |- ( A u. ( _V \ A ) ) = _V

Syntax hints:   = wceq 1483   _Vcvv 3200   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916

This theorem is referenced by:  undif1  4043  dfif4  4101  hashfxnn0  13124  hashfOLD  13126  fullfunfnv  32053  hfext  32290