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Theorem disjdif2 3515
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 3261 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
\  ( A  i^i  B ) )  =  ( A  \  (/) ) )
2 difin 3386 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
3 dif0 3507 . 2  |-  ( A 
\  (/) )  =  A
41, 2, 33eqtr3g 2244 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
\  B )  =  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1363    \ cdif 3140    i^i cin 3142   (/)c0 3436
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-ext 2170
This theorem depends on definitions:  df-bi 117  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-clab 2175  df-cleq 2181  df-clel 2184  df-nfc 2320  df-ral 2472  df-rab 2476  df-v 2753  df-dif 3145  df-in 3149  df-ss 3156  df-nul 3437
This theorem is referenced by:  setsfun0  12515
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