Theorem pltval3

Description: Alternate expression for the "less than" relation. ( dfpss3 analog.) (Contributed by NM, 4-Nov-2011)

	Ref	Expression
Hypotheses	pleval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pleval2.l	⊢ ≤ = ( le ‘ 𝐾 )
	pleval2.s	⊢ < = ( lt ‘ 𝐾 )
Assertion	pltval3	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋 ) ) )

Proof

Step	Hyp	Ref	Expression
1		pleval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
2		pleval2.l	⊢ ≤ = ( le ‘ 𝐾 )
3		pleval2.s	⊢ < = ( lt ‘ 𝐾 )
4	2 3	pltval	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
5	1 2	posref	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 )
6	5	3adant3	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝑋 ≤ 𝑋 )
7		breq1	⊢ ( 𝑋 = 𝑌 → ( 𝑋 ≤ 𝑋 ↔ 𝑌 ≤ 𝑋 ) )
8	6 7	syl5ibcom	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 = 𝑌 → 𝑌 ≤ 𝑋 ) )
9	8	adantr	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 = 𝑌 → 𝑌 ≤ 𝑋 ) )
10	1 2	posasymb	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) ↔ 𝑋 = 𝑌 ) )
11	10	biimpd	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋 ) → 𝑋 = 𝑌 ) )
12	11	expdimp	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑌 ≤ 𝑋 → 𝑋 = 𝑌 ) )
13	9 12	impbid	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 = 𝑌 ↔ 𝑌 ≤ 𝑋 ) )
14	13	necon3abid	⊢ ( ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ≠ 𝑌 ↔ ¬ 𝑌 ≤ 𝑋 ) )
15	14	pm5.32da	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ↔ ( 𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋 ) ) )
16	4 15	bitrd	⊢ ( ( 𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋 ) ) )