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Theorem pleval2i 18423

Description: One direction of pleval2 18424. (Contributed by Mario Carneiro, 8-Feb-2015.)

**Hypotheses**
Ref	Expression
pleval2.b	⊢ 𝐵 = (Base‘𝐾)
pleval2.l	⊢ ≤ = (le‘𝐾)
pleval2.s	⊢ < = (lt‘𝐾)

**Assertion**
Ref	Expression
pleval2i	⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))

**Proof of Theorem pleval2i**
Step	Hyp	Ref	Expression
1		elfvdm 6913	. . . . . . . . 9 ⊢ (𝑋 ∈ (Base‘𝐾) → 𝐾 ∈ dom Base)
2		pleval2.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eleq2s 2878	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → 𝐾 ∈ dom Base)
4	3	adantr 486	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ dom Base)
5		pleval2.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
6		pleval2.s	. . . . . . . . 9 ⊢ < = (lt‘𝐾)
7	5, 6	pltval 18419	. . . . . . . 8 ⊢ ((𝐾 ∈ dom Base ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
8	7	3expb 1138	. . . . . . 7 ⊢ ((𝐾 ∈ dom Base ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
9	4, 8	mpancom 701	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
10	9	biimpar 483	. . . . 5 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)) → 𝑋 < 𝑌)
11	10	expr 462	. . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ≠ 𝑌 → 𝑋 < 𝑌))
12	11	necon1bd 2973	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (¬ 𝑋 < 𝑌 → 𝑋 = 𝑌))
13	12	orrd 877	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))
14	13	ex 418	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∨ wo 861 = wceq 1570 ∈ wcel 2145 ≠ wne 2955 class class class wbr 5103 dom cdm 5655 ‘cfv 6533 Basecbs 17302 lecple 17350 ltcplt 18397

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1828 ax-4 1842 ax-5 1943 ax-6 2000 ax-7 2041 ax-8 2147 ax-9 2155 ax-10 2178 ax-11 2194 ax-12 2213 ax-ext 2732 ax-sep 5251 ax-nul 5263 ax-pr 5398

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1573 df-fal 1583 df-ex 1813 df-nf 1817 df-sb 2100 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-dif 3902 df-un 3904 df-in 3906 df-ss 3916 df-nul 4280 df-if 4483 df-sn 4585 df-pr 4587 df-op 4591 df-uni 4868 df-br 5104 df-opab 5168 df-mpt 5187 df-id 5550 df-xp 5661 df-rel 5662 df-cnv 5663 df-co 5664 df-dm 5665 df-iota 6489 df-fun 6535 df-fv 6541 df-plt 18417

This theorem is used by: pleval2 18424 pospo 18432

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