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 ) ) )