Theorem undif2 1762

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see difdisj 1758). Part of proof of Corollary 6K of [Enderton] p. 144.

Assertion

Ref	Expression
undif2	⊢ (A ∪ (B ∖ A)) = (A ∪ B)

Proof of Theorem undif2

Step	Hyp	Ref	Expression
1		uncom 1604	. 2 ⊢ (A ∪ (B ∖ A)) = ((B ∖ A) ∪ A)
2		undif1 1761	. 2 ⊢ ((B ∖ A) ∪ A) = (B ∪ A)
3		uncom 1604	. 2 ⊢ (B ∪ A) = (A ∪ B)
4	1, 2, 3	3eqtr 1123	1 ⊢ (A ∪ (B ∖ A)) = (A ∪ B)

Syntax hints:   = wceq 1091   ∖ cdif 1484   ∪ cun 1485

This theorem is referenced by:  ssundif 1764  difex2 1951  undom 3342  unfi 3441  kmlem10 3589  infxpidmlem12 4944

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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