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Theorem ltprord 9454
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 2501 . . . . 5  |-  ( x  =  A  ->  (
x  e.  P.  <->  A  e.  P. ) )
21anbi1d 709 . . . 4  |-  ( x  =  A  ->  (
( x  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  y  e.  P. )
) )
3 psseq1 3558 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y
) )
42, 3anbi12d 715 . . 3  |-  ( x  =  A  ->  (
( ( x  e. 
P.  /\  y  e.  P. )  /\  x  C.  y )  <->  ( ( A  e.  P.  /\  y  e.  P. )  /\  A  C.  y ) ) )
5 eleq1 2501 . . . . 5  |-  ( y  =  B  ->  (
y  e.  P.  <->  B  e.  P. ) )
65anbi2d 708 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  B  e.  P. )
) )
7 psseq2 3559 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B
) )
86, 7anbi12d 715 . . 3  |-  ( y  =  B  ->  (
( ( A  e. 
P.  /\  y  e.  P. )  /\  A  C.  y )  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
9 df-ltp 9409 . . 3  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
104, 8, 9brabg 4740 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
1110bianabs 888 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 187    /\ wa 370    = wceq 1437    e. wcel 1870    C. wpss 3443   class class class wbr 4426   P.cnp 9283    <P cltp 9287
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-rab 2791  df-v 3089  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-pss 3458  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-br 4427  df-opab 4485  df-ltp 9409
This theorem is referenced by:  ltsopr  9456  ltaddpr  9458  ltexprlem7  9466  ltexpri  9467  suplem1pr  9476  suplem2pr  9477
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