Theorem pleval2i

Description: One direction of pleval2 . (Contributed by Mario Carneiro, 8-Feb-2015)

	Ref	Expression
Hypotheses	pleval2.b	⊢ 𝐵 = ( Base ‘ 𝐾 )
	pleval2.l	⊢ ≤ = ( le ‘ 𝐾 )
	pleval2.s	⊢ < = ( lt ‘ 𝐾 )
Assertion	pleval2i	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 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		elfvdm	⊢ ( 𝑋 ∈ ( Base ‘ 𝐾 ) → 𝐾 ∈ dom Base )
5	4 1	eleq2s	⊢ ( 𝑋 ∈ 𝐵 → 𝐾 ∈ dom Base )
6	5	adantr	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → 𝐾 ∈ dom Base )
7	2 3	pltval	⊢ ( ( 𝐾 ∈ dom Base ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
8	7	3expb	⊢ ( ( 𝐾 ∈ dom Base ∧ ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
9	6 8	mpancom	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 < 𝑌 ↔ ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) )
10	9	biimpar	⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ ( 𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌 ) ) → 𝑋 < 𝑌 )
11	10	expr	⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 ≠ 𝑌 → 𝑋 < 𝑌 ) )
12	11	necon1bd	⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( ¬ 𝑋 < 𝑌 → 𝑋 = 𝑌 ) )
13	12	orrd	⊢ ( ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) ∧ 𝑋 ≤ 𝑌 ) → ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) )
14	13	ex	⊢ ( ( 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ) → ( 𝑋 ≤ 𝑌 → ( 𝑋 < 𝑌 ∨ 𝑋 = 𝑌 ) ) )