This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem undif1

Description: Absorption of difference by union. This decomposes a union into two disjoint classes (see disjdif ). Theorem 35 of Suppes p. 29. (Contributed by NM, 19-May-1998)

Ref Expression
Assertion undif1 
|- ( ( A \ B ) u. B ) = ( A u. B )

Proof

Step Hyp Ref Expression
1 undir 
 |-  ( ( A i^i ( _V \ B ) ) u. B ) = ( ( A u. B ) i^i ( ( _V \ B ) u. B ) )
2 invdif 
 |-  ( A i^i ( _V \ B ) ) = ( A \ B )
3 2 uneq1i 
 |-  ( ( A i^i ( _V \ B ) ) u. B ) = ( ( A \ B ) u. B )
4 uncom 
 |-  ( ( _V \ B ) u. B ) = ( B u. ( _V \ B ) )
5 unvdif 
 |-  ( B u. ( _V \ B ) ) = _V
6 4 5 eqtri 
 |-  ( ( _V \ B ) u. B ) = _V
7 6 ineq2i 
 |-  ( ( A u. B ) i^i ( ( _V \ B ) u. B ) ) = ( ( A u. B ) i^i _V )
8 inv1 
 |-  ( ( A u. B ) i^i _V ) = ( A u. B )
9 7 8 eqtri 
 |-  ( ( A u. B ) i^i ( ( _V \ B ) u. B ) ) = ( A u. B )
10 1 3 9 3eqtr3i 
 |-  ( ( A \ B ) u. B ) = ( A u. B )