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 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ¬ 𝑋 < 𝑋 )

Proof

Step	Hyp	Ref	Expression
1		pltne.s	⊢ < = ( lt ‘ 𝐾 )
2		eqid	⊢ 𝑋 = 𝑋
3	1	pltne	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 < 𝑋 → 𝑋 ≠ 𝑋 ) )
4	3	3anidm23	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 < 𝑋 → 𝑋 ≠ 𝑋 ) )
5	4	necon2bd	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ( 𝑋 = 𝑋 → ¬ 𝑋 < 𝑋 ) )
6	2 5	mpi	⊢ ( ( 𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ) → ¬ 𝑋 < 𝑋 )