Theorem disjdif2 3570

Description: The difference of a class and a class disjoint from it is the original class. (Contributed by BJ, 21-Apr-2019.)

Assertion

Ref	Expression
disjdif2	|- ( ( A i^i B ) = (/) -> ( A \ B ) = A )

Proof of Theorem disjdif2

Step	Hyp	Ref	Expression
1		difeq2 3316	. 2 |- ( ( A i^i B ) = (/) -> ( A \ ( A i^i B ) ) = ( A \ (/) ) )
2		difin 3441	. 2 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		dif0 3562	. 2 |- ( A \ (/) ) = A
4	1, 2, 3	3eqtr3g 2285	1 |- ( ( A i^i B ) = (/) -> ( A \ B ) = A )

Syntax hints:   -> wi 4   = wceq 1395   \ cdif 3194   i^i cin 3196   (/)c0 3491

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rab 2517  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492

This theorem is referenced by:  opwo0id  4335  setsfun0  13068