Theorem ltprord 9852

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.

Step	Hyp	Ref	Expression
1		eleq1 2689	. . . . 5 |- ( x = A -> ( x e. P. <-> A e. P. ) )
2	1	anbi1d 741	. . . 4 |- ( x = A -> ( ( x e. P. /\ y e. P. ) <-> ( A e. P. /\ y e. P. ) ) )
3		psseq1 3694	. . . 4 |- ( x = A -> ( x C. y <-> A C. y ) )
4	2, 3	anbi12d 747	. . 3 |- ( x = A -> ( ( ( x e. P. /\ y e. P. ) /\ x C. y ) <-> ( ( A e. P. /\ y e. P. ) /\ A C. y ) ) )
5		eleq1 2689	. . . . 5 |- ( y = B -> ( y e. P. <-> B e. P. ) )
6	5	anbi2d 740	. . . 4 |- ( y = B -> ( ( A e. P. /\ y e. P. ) <-> ( A e. P. /\ B e. P. ) ) )
7		psseq2 3695	. . . 4 |- ( y = B -> ( A C. y <-> A C. B ) )
8	6, 7	anbi12d 747	. . 3 |- ( y = B -> ( ( ( A e. P. /\ y e. P. ) /\ A C. y ) <-> ( ( A e. P. /\ B e. P. ) /\ A C. B ) ) )
9		df-ltp 9807	. . 3 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ x C. y ) }
10	4, 8, 9	brabg 4994	. 2 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> ( ( A e. P. /\ B e. P. ) /\ A C. B ) ) )
11	10	bianabs 924	1 |- ( ( A e. P. /\ B e. P. ) -> ( A <P B <-> A C. B ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   C. wpss 3575   class class class wbr 4653   P.cnp 9681   <P cltp 9685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-ltp 9807

This theorem is referenced by:  ltsopr  9854  ltaddpr  9856  ltexprlem7  9864  ltexpri  9865  suplem1pr  9874  suplem2pr  9875