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

Theorem relssi 4794
Description: Inference from subclass principle for relations. (Contributed by NM, 31-Mar-1998.)
Hypotheses
Ref Expression
relssi.1  |-  Rel  A
relssi.2  |-  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
)
Assertion
Ref Expression
relssi  |-  A  C_  B
Distinct variable groups:    x, y, A    x, B, y

Proof of Theorem relssi
StepHypRef Expression
1 relssi.1 . . 3  |-  Rel  A
2 ssrel 4792 . . 3  |-  ( Rel 
A  ->  ( A  C_  B  <->  A. x A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
) ) )
31, 2ax-mp 8 . 2  |-  ( A 
C_  B  <->  A. x A. y ( <. x ,  y >.  e.  A  -> 
<. x ,  y >.  e.  B ) )
4 relssi.2 . . 3  |-  ( <.
x ,  y >.  e.  A  ->  <. x ,  y >.  e.  B
)
54ax-gen 1536 . 2  |-  A. y
( <. x ,  y
>.  e.  A  ->  <. x ,  y >.  e.  B
)
63, 5mpgbir 1540 1  |-  A  C_  B
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176   A.wal 1530    e. wcel 1696    C_ wss 3165   <.cop 3656   Rel wrel 4710
This theorem is referenced by:  xpsspwOLD  4814  resiexg  5013  oprssdm  6018  dftpos4  6269  enssdom  6902  idssen  6922  txuni2  17276  txpss3v  24489  pprodss4v  24495  aoprssdm  28170
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-opab 4094  df-xp 4711  df-rel 4712
  Copyright terms: Public domain W3C validator