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Theorem ssopab2i 4042
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 4040 . 2  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )
2 ssopab2i.1 . . 3  |-  ( ph  ->  ps )
32ax-gen 1354 . 2  |-  A. y
( ph  ->  ps )
41, 3mpg 1356 1  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    C_ wss 2945   {copab 3845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-in 2952  df-ss 2959  df-opab 3847
This theorem is referenced by:  brab2a  4421  opabssxp  4442  funopab4  4965  ssoprab2i  5621  npsspw  6627
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