Theorem undifv 2343

Description: The union of a class and its complement is the universe. Theorem 5.1(5) of [Stoll] p. 17.

Assertion

Ref	Expression
undifv	|- (A u. (V \ A)) = V

Proof of Theorem undifv

Step	Hyp	Ref	Expression
1		dfun3 2249	. 2 |- (A u. (V \ A)) = (V \ ((V \ A) i^i (V \ (V \ A))))
2		difdisj 2341	. . 3 |- ((V \ A) i^i (V \ (V \ A))) = (/)
3	2	difeq2i 2159	. 2 |- (V \ ((V \ A) i^i (V \ (V \ A)))) = (V \ (/))
4		dif0 2339	. 2 |- (V \ (/)) = V
5	1, 3, 4	3eqtr 1502	1 |- (A u. (V \ A)) = V

Syntax hints:   = wceq 958  Vcvv 1814   \ cdif 2047   u. cun 2048   i^i cin 2049  (/)c0 2283

This theorem is referenced by:  undif1 2344

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284

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