Theorem disjdif2 3380

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 3127	. 2 |- ( ( A i^i B ) = (/) -> ( A \ ( A i^i B ) ) = ( A \ (/) ) )
2		difin 3252	. 2 |- ( A \ ( A i^i B ) ) = ( A \ B )
3		dif0 3372	. 2 |- ( A \ (/) ) = A
4	1, 2, 3	3eqtr3g 2150	1 |- ( ( A i^i B ) = (/) -> ( A \ B ) = A )

Syntax hints:   -> wi 4   = wceq 1296   \ cdif 3010   i^i cin 3012   (/)c0 3302

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077

This theorem depends on definitions:  df-bi 116  df-tru 1299  df-fal 1302  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rab 2379  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-nul 3303

This theorem is referenced by:  setsfun0  11679