Theorem reldif 4654

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

Step	Hyp	Ref	Expression
1		difss 3197	. 2 |- ( A \ B ) C_ A
2		relss 4621	. 2 |- ( ( A \ B ) C_ A -> ( Rel A -> Rel ( A \ B ) ) )
3	1, 2	ax-mp 5	1 |- ( Rel A -> Rel ( A \ B ) )

Syntax hints:   -> wi 4   \ cdif 3063   C_ wss 3066   Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-rel 4541

This theorem is referenced by:  difopab  4667