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Metamath Proof Explorer

Theorem 2llnne2N

Description: Condition implying that two intersecting lines are different. (Contributed by NM, 13-Jun-2012) (New usage is discouraged.)

Ref Expression
Hypotheses 2lnne.l 
|- .<_ = ( le ` K )
2lnne.j 
|- .\/ = ( join ` K )
2lnne.a 
|- A = ( Atoms ` K )
Assertion 2llnne2N 
|- ( ( K e. HL /\ ( P e. A /\ R e. A ) /\ -. P .<_ ( R .\/ Q ) ) -> ( R .\/ P ) =/= ( R .\/ Q ) )

Proof

Step Hyp Ref Expression
1 2lnne.l 
 |-  .<_ = ( le ` K )
2 2lnne.j 
 |-  .\/ = ( join ` K )
3 2lnne.a 
 |-  A = ( Atoms ` K )
4 simpl 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) ) -> K e. HL )
5 simprr 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) ) -> R e. A )
6 simprl 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) ) -> P e. A )
7 1 2 3 hlatlej2 
 |-  ( ( K e. HL /\ R e. A /\ P e. A ) -> P .<_ ( R .\/ P ) )
8 4 5 6 7 syl3anc 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) ) -> P .<_ ( R .\/ P ) )
9 breq2 
 |-  ( ( R .\/ P ) = ( R .\/ Q ) -> ( P .<_ ( R .\/ P ) <-> P .<_ ( R .\/ Q ) ) )
10 8 9 syl5ibcom 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) ) -> ( ( R .\/ P ) = ( R .\/ Q ) -> P .<_ ( R .\/ Q ) ) )
11 10 necon3bd 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) ) -> ( -. P .<_ ( R .\/ Q ) -> ( R .\/ P ) =/= ( R .\/ Q ) ) )
12 11 3impia 
 |-  ( ( K e. HL /\ ( P e. A /\ R e. A ) /\ -. P .<_ ( R .\/ Q ) ) -> ( R .\/ P ) =/= ( R .\/ Q ) )