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Theorem ltprord 9420
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 2539 . . . . 5  |-  ( x  =  A  ->  (
x  e.  P.  <->  A  e.  P. ) )
21anbi1d 704 . . . 4  |-  ( x  =  A  ->  (
( x  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  y  e.  P. )
) )
3 psseq1 3596 . . . 4  |-  ( x  =  A  ->  (
x  C.  y  <->  A  C.  y
) )
42, 3anbi12d 710 . . 3  |-  ( x  =  A  ->  (
( ( x  e. 
P.  /\  y  e.  P. )  /\  x  C.  y )  <->  ( ( A  e.  P.  /\  y  e.  P. )  /\  A  C.  y ) ) )
5 eleq1 2539 . . . . 5  |-  ( y  =  B  ->  (
y  e.  P.  <->  B  e.  P. ) )
65anbi2d 703 . . . 4  |-  ( y  =  B  ->  (
( A  e.  P.  /\  y  e.  P. )  <->  ( A  e.  P.  /\  B  e.  P. )
) )
7 psseq2 3597 . . . 4  |-  ( y  =  B  ->  ( A  C.  y  <->  A  C.  B
) )
86, 7anbi12d 710 . . 3  |-  ( y  =  B  ->  (
( ( A  e. 
P.  /\  y  e.  P. )  /\  A  C.  y )  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
9 df-ltp 9375 . . 3  |-  <P  =  { <. x ,  y
>.  |  ( (
x  e.  P.  /\  y  e.  P. )  /\  x  C.  y ) }
104, 8, 9brabg 4772 . 2  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  ( ( A  e.  P.  /\  B  e.  P. )  /\  A  C.  B ) ) )
1110bianabs 878 1  |-  ( ( A  e.  P.  /\  B  e.  P. )  ->  ( A  <P  B  <->  A  C.  B
) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767    C. wpss 3482   class class class wbr 4453   P.cnp 9249    <P cltp 9253
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pr 4692
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-rab 2826  df-v 3120  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-pss 3497  df-nul 3791  df-if 3946  df-sn 4034  df-pr 4036  df-op 4040  df-br 4454  df-opab 4512  df-ltp 9375
This theorem is referenced by:  ltsopr  9422  ltaddpr  9424  ltexprlem7  9432  ltexpri  9433  suplem1pr  9442  suplem2pr  9443
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