Theorem reldif 4853

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 3335	. 2 |- ( A \ B ) C_ A
2		relss 4819	. 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 3198   C_ wss 3201   Rel wrel 4736

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-rel 4738

This theorem is referenced by:  difopab  4869  fundif  5381