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Theorem ltprord 8801
Description: Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
ltprord  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B ) )

Proof of Theorem ltprord
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2426 . . . . 5  |-  ( x  =  A  ->  (
x  e.  P.  <->  A  e.  P. ) )
21anbi1d 685 . . . 4  |-  ( x  =  A  ->  (
( x  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  y  e.  P. )
) )
3 psseq1 3350 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y ) )
42, 3anbi12d 691 . . 3  |-  ( x  =  A  ->  (
( ( x  e. 
P.  /\  y  e.  P. )  /\  x  C.  y )  <->  ( ( A  e.  P.  /\  y  e.  P. )  /\  A  C.  y ) ) )
5 eleq1 2426 . . . . 5  |-  ( y  =  B  ->  (
y  e.  P.  <->  B  e.  P. ) )
65anbi2d 684 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  B  e.  P. )
) )
7 psseq2 3351 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B ) )
86, 7anbi12d 691 . . 3  |-  ( y  =  B  ->  (
( ( A  e. 
P.  /\  y  e.  P. )  /\  A  C.  y )  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
9 df-ltp 8756 . . 3  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
104, 8, 9brabg 4387 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
1110bianabs 850 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 176    /\ wa 358    = wceq 1647    e. wcel 1715    C. wpss 3239   class class class wbr 4125   P.cnp 8628    <P cltp 8632
This theorem is referenced by:  ltsopr  8803  ltaddpr  8805  ltexprlem7  8813  ltexpri  8814  suplem1pr  8823  suplem2pr  8824
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1551  ax-5 1562  ax-17 1621  ax-9 1659  ax-8 1680  ax-14 1719  ax-6 1734  ax-7 1739  ax-11 1751  ax-12 1937  ax-ext 2347  ax-sep 4243  ax-nul 4251  ax-pr 4316
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 937  df-tru 1324  df-ex 1547  df-nf 1550  df-sb 1654  df-eu 2221  df-mo 2222  df-clab 2353  df-cleq 2359  df-clel 2362  df-nfc 2491  df-ne 2531  df-rab 2637  df-v 2875  df-dif 3241  df-un 3243  df-in 3245  df-ss 3252  df-pss 3254  df-nul 3544  df-if 3655  df-sn 3735  df-pr 3736  df-op 3738  df-br 4126  df-opab 4180  df-ltp 8756
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