This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

Theorem pleval2

Description: "Less than or equal to" in terms of "less than". ( sspss analog.) (Contributed by NM, 17-Oct-2011) (Revised by Mario Carneiro, 8-Feb-2015)

Ref Expression
Hypotheses pleval2.b 
|- B = ( Base ` K )
pleval2.l 
|- .<_ = ( le ` K )
pleval2.s 
|- .< = ( lt ` K )
Assertion pleval2 
|- ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y <-> ( X .< Y \/ X = Y ) ) )

Proof

Step Hyp Ref Expression
1 pleval2.b 
 |-  B = ( Base ` K )
2 pleval2.l 
 |-  .<_ = ( le ` K )
3 pleval2.s 
 |-  .< = ( lt ` K )
4 1 2 3 pleval2i 
 |-  ( ( X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )
5 4 3adant1 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y -> ( X .< Y \/ X = Y ) ) )
6 2 3 pltle 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .< Y -> X .<_ Y ) )
7 1 2 posref 
 |-  ( ( K e. Poset /\ X e. B ) -> X .<_ X )
8 7 3adant3 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> X .<_ X )
9 breq2 
 |-  ( X = Y -> ( X .<_ X <-> X .<_ Y ) )
10 8 9 syl5ibcom 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X = Y -> X .<_ Y ) )
11 6 10 jaod 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( ( X .< Y \/ X = Y ) -> X .<_ Y ) )
12 5 11 impbid 
 |-  ( ( K e. Poset /\ X e. B /\ Y e. B ) -> ( X .<_ Y <-> ( X .< Y \/ X = Y ) ) )