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Theorem disjdif2 3875
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 3578 . 2  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
\  ( A  i^i  B ) )  =  ( A  \  (/) ) )
2 difin 3711 . 2  |-  ( A 
\  ( A  i^i  B ) )  =  ( A  \  B )
3 dif0 3866 . 2  |-  ( A 
\  (/) )  =  A
41, 2, 33eqtr3g 2487 1  |-  ( ( A  i^i  B )  =  (/)  ->  ( A 
\  B )  =  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    \ cdif 3434    i^i cin 3436   (/)c0 3762
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rab 2785  df-v 3084  df-dif 3440  df-in 3444  df-ss 3451  df-nul 3763
This theorem is referenced by:  ptbasfi  20588  fzdif2  28369  bj-2upln1upl  31580  dvmptfprodlem  37683
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