Theorem pltirr

Description: The "less than" relation is not reflexive. ( pssirr analog.) (Contributed by NM, 7-Feb-2012)

	Ref	Expression
Hypothesis	pltne.s	|- .< = ( lt ` K )
Assertion	pltirr	|- ( ( K e. A /\ X e. B ) -> -. X .< X )

Proof

Step	Hyp	Ref	Expression
1		pltne.s	|- .< = ( lt ` K )
2		eqid	|- X = X
3	1	pltne	|- ( ( K e. A /\ X e. B /\ X e. B ) -> ( X .< X -> X =/= X ) )
4	3	3anidm23	|- ( ( K e. A /\ X e. B ) -> ( X .< X -> X =/= X ) )
5	4	necon2bd	|- ( ( K e. A /\ X e. B ) -> ( X = X -> -. X .< X ) )
6	2 5	mpi	|- ( ( K e. A /\ X e. B ) -> -. X .< X )