Theorem relss 4673

Description: Subclass theorem for relation predicate. Theorem 2 of [Suppes] p. 58. (Contributed by NM, 15-Aug-1994.)

Assertion

Ref	Expression
relss	|- ( A C_ B -> ( Rel B -> Rel A ) )

Proof of Theorem relss

Step	Hyp	Ref	Expression
1		sstr2 3135	. 2 |- ( A C_ B -> ( B C_ ( _V X. _V ) -> A C_ ( _V X. _V ) ) )
2		df-rel 4593	. 2 |- ( Rel B <-> B C_ ( _V X. _V ) )
3		df-rel 4593	. 2 |- ( Rel A <-> A C_ ( _V X. _V ) )
4	1, 2, 3	3imtr4g 204	1 |- ( A C_ B -> ( Rel B -> Rel A ) )

Syntax hints:   -> wi 4   _Vcvv 2712   C_ wss 3102   X. cxp 4584   Rel wrel 4591

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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139

This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115  df-rel 4593

This theorem is referenced by:  relin1  4704  relin2  4705  reldif  4706  relres  4894  iss  4912  cnvdif  4992  funss  5189  funssres  5212  fliftcnv  5745  fliftfun  5746  reltpos  6197  tpostpos  6211  swoer  6508  erinxp  6554  ltrel  7939  lerel  7941  txdis1cn  12678  xmeter  12836