Theorem relss 4586

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 3070	. 2 |- ( A C_ B -> ( B C_ ( _V X. _V ) -> A C_ ( _V X. _V ) ) )
2		df-rel 4506	. 2 |- ( Rel B <-> B C_ ( _V X. _V ) )
3		df-rel 4506	. 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 2657   C_ wss 3037   X. cxp 4497   Rel wrel 4504

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-11 1467  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097

This theorem depends on definitions:  df-bi 116  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-in 3043  df-ss 3050  df-rel 4506

This theorem is referenced by:  relin1  4617  relin2  4618  reldif  4619  relres  4805  iss  4823  cnvdif  4903  funss  5100  funssres  5123  fliftcnv  5650  fliftfun  5651  reltpos  6101  tpostpos  6115  swoer  6411  erinxp  6457  ltrel  7750  lerel  7752  txdis1cn  12289  xmeter  12425