MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssbrd Structured version   Unicode version

Theorem ssbrd 4253
Description: Deduction from a subclass relationship of binary relations. (Contributed by NM, 30-Apr-2004.)
Hypothesis
Ref Expression
ssbrd.1  |-  ( ph  ->  A  C_  B )
Assertion
Ref Expression
ssbrd  |-  ( ph  ->  ( C A D  ->  C B D ) )

Proof of Theorem ssbrd
StepHypRef Expression
1 ssbrd.1 . . 3  |-  ( ph  ->  A  C_  B )
21sseld 3347 . 2  |-  ( ph  ->  ( <. C ,  D >.  e.  A  ->  <. C ,  D >.  e.  B ) )
3 df-br 4213 . 2  |-  ( C A D  <->  <. C ,  D >.  e.  A )
4 df-br 4213 . 2  |-  ( C B D  <->  <. C ,  D >.  e.  B )
52, 3, 43imtr4g 262 1  |-  ( ph  ->  ( C A D  ->  C B D ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    e. wcel 1725    C_ wss 3320   <.cop 3817   class class class wbr 4212
This theorem is referenced by:  ssbri  4254  sess1  4550  brrelex12  4915  coss1  5028  coss2  5029  eqbrrdva  5042  ersym  6917  ertr  6920  fpwwe2lem6  8510  fpwwe2lem7  8511  fpwwe2lem9  8513  fpwwe2lem12  8516  fpwwe2lem13  8517  fpwwe2  8518  fthres2  14129  invfuc  14171  pospo  14430  dirref  14680  efgcpbl  15388  frgpuplem  15404  subrguss  15883  znleval  16835  ustref  18248  ustuqtop4  18274  isucn2  18309  metider  24289  fundmpss  25390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-in 3327  df-ss 3334  df-br 4213
  Copyright terms: Public domain W3C validator