Theorem disjdif2 3446

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 3193	. 2 |- ( ( A i^i B ) = (/) -> ( A \ ( A i^i B ) ) = ( A \ (/) ) )
2		difin 3318	. 2 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		dif0 3438	. 2 |- ( A \ (/) ) = A
4	1, 2, 3	3eqtr3g 2196	1 |- ( ( A i^i B ) = (/) -> ( A \ B ) = A )

Syntax hints:   -> wi 4   = wceq 1332   \ cdif 3073   i^i cin 3075   (/)c0 3368

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 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122

This theorem depends on definitions:  df-bi 116  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rab 2426  df-v 2691  df-dif 3078  df-in 3082  df-ss 3089  df-nul 3369

This theorem is referenced by:  setsfun0  12034