Theorem undifv 1760

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 ∪ (V ∖ A)) = V

Proof of Theorem undifv

Step	Hyp	Ref	Expression
1		dfun3 1671	. 2 ⊢ (A ∪ (V ∖ A)) = (V ∖ ((V ∖ A) ∩ (V ∖ (V ∖ A))))
2		difdisj 1758	. . 3 ⊢ ((V ∖ A) ∩ (V ∖ (V ∖ A))) = ∅
3	2	difeq2i 1585	. 2 ⊢ (V ∖ ((V ∖ A) ∩ (V ∖ (V ∖ A)))) = (V ∖ ∅)
4		dif0 1756	. 2 ⊢ (V ∖ ∅) = V
5	1, 3, 4	3eqtr 1123	1 ⊢ (A ∪ (V ∖ A)) = V

Syntax hints:   = wceq 1091  Vcvv 1348   ∖ cdif 1484   ∪ cun 1485   ∩ cin 1486  ∅c0 1707

This theorem is referenced by:  undif1 1761

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708

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