Theorem undif1 1761

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see difdisj 1758). Theorem 35 of [Suppes] p. 29.

Assertion

Ref	Expression
undif1	⊢ ((A ∖ B) ∪ B) = (A ∪ B)

Proof of Theorem undif1

Step	Hyp	Ref	Expression
1		invdif 1674	. . 3 ⊢ (A ∩ (V ∖ B)) = (A ∖ B)
2	1	uneq1i 1607	. 2 ⊢ ((A ∩ (V ∖ B)) ∪ B) = ((A ∖ B) ∪ B)
3		undir 1679	. . 3 ⊢ ((A ∩ (V ∖ B)) ∪ B) = ((A ∪ B) ∩ ((V ∖ B) ∪ B))
4		uncom 1604	. . . . 5 ⊢ ((V ∖ B) ∪ B) = (B ∪ (V ∖ B))
5		undifv 1760	. . . . 5 ⊢ (B ∪ (V ∖ B)) = V
6	4, 5	eqtr 1119	. . . 4 ⊢ ((V ∖ B) ∪ B) = V
7	6	ineq2i 1642	. . 3 ⊢ ((A ∪ B) ∩ ((V ∖ B) ∪ B)) = ((A ∪ B) ∩ V)
8		inv 1723	. . 3 ⊢ ((A ∪ B) ∩ V) = (A ∪ B)
9	3, 7, 8	3eqtr 1123	. 2 ⊢ ((A ∩ (V ∖ B)) ∪ B) = (A ∪ B)
10	2, 9	eqtr3 1121	1 ⊢ ((A ∖ B) ∪ B) = (A ∪ B)

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

This theorem is referenced by:  undif2 1762  unidif0 1944  infdif 4948

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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