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Theorem relss 4513
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
StepHypRef Expression
1 sstr2 3030 . 2  |-  ( A 
C_  B  ->  ( B  C_  ( _V  X.  _V )  ->  A  C_  ( _V  X.  _V )
) )
2 df-rel 4435 . 2  |-  ( Rel 
B  <->  B  C_  ( _V 
X.  _V ) )
3 df-rel 4435 . 2  |-  ( Rel 
A  <->  A  C_  ( _V 
X.  _V ) )
41, 2, 33imtr4g 203 1  |-  ( A 
C_  B  ->  ( Rel  B  ->  Rel  A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4   _Vcvv 2619    C_ wss 2997    X. cxp 4426   Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-rel 4435
This theorem is referenced by:  relin1  4543  relin2  4544  reldif  4545  relres  4728  iss  4745  cnvdif  4825  funss  5020  funssres  5042  fliftcnv  5556  fliftfun  5557  reltpos  5997  tpostpos  6011  swoer  6300  erinxp  6346  ltrel  7527  lerel  7529
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