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Metamath Proof Explorer

Theorem ltprord

Description: Positive real 'less than' in terms of proper subset. (Contributed by NM, 20-Feb-1996) (New usage is discouraged.)

Ref Expression
Assertion ltprord 
|- ( ( A e. P. /\ B e. P. ) -> ( A 
 A C. B ) )

Proof

Step Hyp Ref Expression
1 eleq1 
 |-  ( x = A -> ( x e. P. <-> A e. P. ) )
2 1 anbi1d 
 |-  ( x = A -> ( ( x e. P. /\ y e. P. ) <-> ( A e. P. /\ y e. P. ) ) )
3 psseq1 
 |-  ( x = A -> ( x C. y <-> A C. y ) )
4 2 3 anbi12d 
 |-  ( x = A -> ( ( ( x e. P. /\ y e. P. ) /\ x C. y ) <-> ( ( A e. P. /\ y e. P. ) /\ A C. y ) ) )
5 eleq1 
 |-  ( y = B -> ( y e. P. <-> B e. P. ) )
6 5 anbi2d 
 |-  ( y = B -> ( ( A e. P. /\ y e. P. ) <-> ( A e. P. /\ B e. P. ) ) )
7 psseq2 
 |-  ( y = B -> ( A C. y <-> A C. B ) )
8 6 7 anbi12d 
 |-  ( y = B -> ( ( ( A e. P. /\ y e. P. ) /\ A C. y ) <-> ( ( A e. P. /\ B e. P. ) /\ A C. B ) ) )
9 df-ltp 
 |-  
. | ( ( x e. P. /\ y e. P. ) /\ x C. y ) }
10 4 8 9 brabg 
 |-  ( ( A e. P. /\ B e. P. ) -> ( A 
 ( ( A e. P. /\ B e. P. ) /\ A C. B ) ) )
11 10 bianabs 
 |-  ( ( A e. P. /\ B e. P. ) -> ( A 
 A C. B ) )