Theorem disjdif2 3472

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 3219	. 2 |- ( ( A i^i B ) = (/) -> ( A \ ( A i^i B ) ) = ( A \ (/) ) )
2		difin 3344	. 2 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		dif0 3464	. 2 |- ( A \ (/) ) = A
4	1, 2, 3	3eqtr3g 2213	1 |- ( ( A i^i B ) = (/) -> ( A \ B ) = A )

Syntax hints:   -> wi 4   = wceq 1335   \ cdif 3099   i^i cin 3101   (/)c0 3394

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 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rab 2444  df-v 2714  df-dif 3104  df-in 3108  df-ss 3115  df-nul 3395

This theorem is referenced by:  setsfun0  12237