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Theorem oplem1 962
Description: A specialized lemma for set theory (ordered pair theorem). (Contributed by NM, 18-Oct-1995.) (Proof shortened by Wolf Lammen, 8-Dec-2012.)
Hypotheses
Ref Expression
oplem1.1  |-  ( ph  ->  ( ps  \/  ch ) )
oplem1.2  |-  ( ph  ->  ( th  \/  ta ) )
oplem1.3  |-  ( ps  <->  th )
oplem1.4  |-  ( ch 
->  ( th  <->  ta )
)
Assertion
Ref Expression
oplem1  |-  ( ph  ->  ps )

Proof of Theorem oplem1
StepHypRef Expression
1 oplem1.3 . . . . . . 7  |-  ( ps  <->  th )
21notbii 296 . . . . . 6  |-  ( -. 
ps 
<->  -.  th )
3 oplem1.1 . . . . . . 7  |-  ( ph  ->  ( ps  \/  ch ) )
43ord 377 . . . . . 6  |-  ( ph  ->  ( -.  ps  ->  ch ) )
52, 4syl5bir 218 . . . . 5  |-  ( ph  ->  ( -.  th  ->  ch ) )
6 oplem1.2 . . . . . 6  |-  ( ph  ->  ( th  \/  ta ) )
76ord 377 . . . . 5  |-  ( ph  ->  ( -.  th  ->  ta ) )
85, 7jcad 533 . . . 4  |-  ( ph  ->  ( -.  th  ->  ( ch  /\  ta )
) )
9 oplem1.4 . . . . 5  |-  ( ch 
->  ( th  <->  ta )
)
109biimpar 485 . . . 4  |-  ( ( ch  /\  ta )  ->  th )
118, 10syl6 33 . . 3  |-  ( ph  ->  ( -.  th  ->  th ) )
1211pm2.18d 111 . 2  |-  ( ph  ->  th )
1312, 1sylibr 212 1  |-  ( ph  ->  ps )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371
This theorem is referenced by:  preqr1  4206
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