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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
StepHypRef 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
41, 2, 33eqtr3g 2150 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
\  B )  =  A )
Colors of variables: wff set class
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
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