Theorem reldif 4847

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 3333	. 2 |- ( A \ B ) C_ A
2		relss 4813	. 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 3197   C_ wss 3200   Rel wrel 4730

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-rel 4732

This theorem is referenced by:  difopab  4863  fundif  5374