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Theorem reldif 4742
Description: A difference cutting down a relation is a relation. (Contributed by NM, 31-Mar-1998.)
Assertion
Ref Expression
reldif  |-  ( Rel 
A  ->  Rel  ( A 
\  B ) )

Proof of Theorem reldif
StepHypRef Expression
1 difss 3261 . 2  |-  ( A 
\  B )  C_  A
2 relss 4709 . 2  |-  ( ( A  \  B ) 
C_  A  ->  ( Rel  A  ->  Rel  ( A 
\  B ) ) )
31, 2ax-mp 5 1  |-  ( Rel 
A  ->  Rel  ( A 
\  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \ cdif 3126    C_ wss 3129   Rel wrel 4627
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-dif 3131  df-in 3135  df-ss 3142  df-rel 4629
This theorem is referenced by:  difopab  4755
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