Theorem disjdif2 3436

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 3183	. 2 |- ( ( A i^i B ) = (/) -> ( A \ ( A i^i B ) ) = ( A \ (/) ) )
2		difin 3308	. 2 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		dif0 3428	. 2 |- ( A \ (/) ) = A
4	1, 2, 3	3eqtr3g 2193	1 |- ( ( A i^i B ) = (/) -> ( A \ B ) = A )

Syntax hints:   -> wi 4   = wceq 1331   \ cdif 3063   i^i cin 3065   (/)c0 3358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rab 2423  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359

This theorem is referenced by:  setsfun0  11984