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Theorem pleval2i 18355

Description: One direction of pleval2 18356. (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 6924	. . . . . . . . 9 ⊢ (𝑋 ∈ (Base‘𝐾) → 𝐾 ∈ dom Base)
2		pleval2.b	. . . . . . . . 9 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eleq2s 2851	. . . . . . . 8 ⊢ (𝑋 ∈ 𝐵 → 𝐾 ∈ dom Base)
4	3	adantr 480	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → 𝐾 ∈ dom Base)
5		pleval2.l	. . . . . . . . 9 ⊢ ≤ = (le‘𝐾)
6		pleval2.s	. . . . . . . . 9 ⊢ < = (lt‘𝐾)
7	5, 6	pltval 18351	. . . . . . . 8 ⊢ ((𝐾 ∈ dom Base ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
8	7	3expb 1120	. . . . . . 7 ⊢ ((𝐾 ∈ dom Base ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵)) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
9	4, 8	mpancom 688	. . . . . 6 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
10	9	biimpar 477	. . . . 5 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)) → 𝑋 < 𝑌)
11	10	expr 456	. . . 4 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ≠ 𝑌 → 𝑋 < 𝑌))
12	11	necon1bd 2949	. . 3 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (¬ 𝑋 < 𝑌 → 𝑋 = 𝑌))
13	12	orrd 863	. 2 ⊢ (((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))
14	13	ex 412	1 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 → (𝑋 < 𝑌 ∨ 𝑋 = 𝑌)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ≠ wne 2931 class class class wbr 5125 dom cdm 5667 ‘cfv 6542 Basecbs 17230 lecple 17284 ltcplt 18329

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2706 ax-sep 5278 ax-nul 5288 ax-pr 5414

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2726 df-clel 2808 df-nfc 2884 df-ne 2932 df-ral 3051 df-rex 3060 df-rab 3421 df-v 3466 df-dif 3936 df-un 3938 df-in 3940 df-ss 3950 df-nul 4316 df-if 4508 df-sn 4609 df-pr 4611 df-op 4615 df-uni 4890 df-br 5126 df-opab 5188 df-mpt 5208 df-id 5560 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-iota 6495 df-fun 6544 df-fv 6550 df-plt 18349

This theorem is referenced by: pleval2 18356 pospo 18364

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