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Theorem ltprord 8899
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 2495 . . . . 5  |-  ( x  =  A  ->  (
x  e.  P.  <->  A  e.  P. ) )
21anbi1d 686 . . . 4  |-  ( x  =  A  ->  (
( x  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  y  e.  P. )
) )
3 psseq1 3426 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y ) )
42, 3anbi12d 692 . . 3  |-  ( x  =  A  ->  (
( ( x  e. 
P.  /\  y  e.  P. )  /\  x  C.  y )  <->  ( ( A  e.  P.  /\  y  e.  P. )  /\  A  C.  y ) ) )
5 eleq1 2495 . . . . 5  |-  ( y  =  B  ->  (
y  e.  P.  <->  B  e.  P. ) )
65anbi2d 685 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  B  e.  P. )
) )
7 psseq2 3427 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B ) )
86, 7anbi12d 692 . . 3  |-  ( y  =  B  ->  (
( ( A  e. 
P.  /\  y  e.  P. )  /\  A  C.  y )  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
9 df-ltp 8854 . . 3  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
104, 8, 9brabg 4466 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
1110bianabs 851 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    <-> wb 177    /\ wa 359    = wceq 1652    e. wcel 1725    C. wpss 3313   class class class wbr 4204   P.cnp 8726    <P cltp 8730
This theorem is referenced by:  ltsopr  8901  ltaddpr  8903  ltexprlem7  8911  ltexpri  8912  suplem1pr  8921  suplem2pr  8922
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-pss 3328  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-ltp 8854
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