MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ssopab2i Structured version   Unicode version

Theorem ssopab2i 4474
Description: Inference of ordered pair abstraction subclass from implication. (Contributed by NM, 5-Apr-1995.)
Hypothesis
Ref Expression
ssopab2i.1  |-  ( ph  ->  ps )
Assertion
Ref Expression
ssopab2i  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps }

Proof of Theorem ssopab2i
StepHypRef Expression
1 ssopab2 4472 . 2  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )
2 ssopab2i.1 . . 3  |-  ( ph  ->  ps )
32ax-gen 1555 . 2  |-  A. y
( ph  ->  ps )
41, 3mpg 1557 1  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1549    C_ wss 3312   {copab 4257
This theorem is referenced by:  brab2a  4919  opabssxp  4942  funopab4  5480  ssoprab2i  6154  cardf2  7822  dfac3  7994  axdc2lem  8320  fpwwe2lem1  8498  canthwe  8518  fullfunc  14095  fthfunc  14096  isfull  14099  isfth  14103  ipoval  14572  ipolerval  14574  eqgfval  14980  2ndcctbss  17510  nvss  22064  ajfval  22302  cvmlift2lem12  24993  relopabVD  28940  dicval  31901
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 2416
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-in 3319  df-ss 3326  df-opab 4259
  Copyright terms: Public domain W3C validator