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Theorem ssopab2i 4292
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 4290 . 2  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )
2 ssopab2i.1 . . 3  |-  ( ph  ->  ps )
32ax-gen 1533 . 2  |-  A. y
( ph  ->  ps )
41, 3mpg 1535 1  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1527    C_ wss 3152   {copab 4076
This theorem is referenced by:  brab2a  4738  opabssxp  4762  funopab4  5289  ssoprab2i  5936  cardf2  7576  dfac3  7748  axdc2lem  8074  fpwwe2lem1  8253  canthwe  8273  fullfunc  13780  fthfunc  13781  isfull  13784  isfth  13788  ipoval  14257  ipolerval  14259  eqgfval  14665  2ndcctbss  17181  nvss  21149  ajfval  21387  cvmlift2lem12  23845  inposet  25278  relopabVD  28677  dicval  31366
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-in 3159  df-ss 3166  df-opab 4078
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