ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssopab2i Unicode version

Theorem ssopab2i 4199
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 4197 . 2  |-  ( A. x A. y ( ph  ->  ps )  ->  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps } )
2 ssopab2i.1 . . 3  |-  ( ph  ->  ps )
32ax-gen 1425 . 2  |-  A. y
( ph  ->  ps )
41, 3mpg 1427 1  |-  { <. x ,  y >.  |  ph }  C_  { <. x ,  y >.  |  ps }
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1329    C_ wss 3071   {copab 3988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990
This theorem is referenced by:  brab2a  4592  opabssxp  4613  funopab4  5160  ssoprab2i  5860  npsspw  7286
  Copyright terms: Public domain W3C validator