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Theorem ssopab2i 4416
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 4414 . 2  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )
2 ssopab2i.1 . . 3  |-  ( ph  ->  ps )
32ax-gen 1552 . 2  |-  A. y
( ph  ->  ps )
41, 3mpg 1554 1  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1546    C_ wss 3256   {copab 4199
This theorem is referenced by:  brab2a  4860  opabssxp  4883  funopab4  5421  ssoprab2i  6094  cardf2  7756  dfac3  7928  axdc2lem  8254  fpwwe2lem1  8432  canthwe  8452  fullfunc  14023  fthfunc  14024  isfull  14027  isfth  14031  ipoval  14500  ipolerval  14502  eqgfval  14908  2ndcctbss  17432  nvss  21913  ajfval  22151  cvmlift2lem12  24773  relopabVD  28347  dicval  31342
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2361
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2367  df-cleq 2373  df-clel 2376  df-nfc 2505  df-in 3263  df-ss 3270  df-opab 4201
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