pltn2lp - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > pltn2lp		Structured version Visualization version GIF version

Theorem pltn2lp 16792

Description: The less-than relation has no 2-cycle loops. (pssn2lp 3670 analog.) (Contributed by NM, 2-Dec-2011.)

**Hypotheses**
Ref	Expression
pltnlt.b	⊢ 𝐵 = (Base‘𝐾)
pltnlt.s	⊢ < = (lt‘𝐾)

**Assertion**
Ref	Expression
pltn2lp	⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋))

**Proof of Theorem pltn2lp**
Step	Hyp	Ref	Expression
1		pltnlt.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐾)
2		eqid 2610	. . . . 5 ⊢ (le‘𝐾) = (le‘𝐾)
3		pltnlt.s	. . . . 5 ⊢ < = (lt‘𝐾)
4	1, 2, 3	pltnle 16789	. . . 4 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌(le‘𝐾)𝑋)
5	4	ex 449	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌(le‘𝐾)𝑋))
6	2, 3	pltle 16784	. . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 < 𝑋 → 𝑌(le‘𝐾)𝑋))
7	6	3com23 1263	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 < 𝑋 → 𝑌(le‘𝐾)𝑋))
8	5, 7	nsyld 153	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 < 𝑋))
9		imnan 437	. 2 ⊢ ((𝑋 < 𝑌 → ¬ 𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋))
10	8, 9	sylib 207	1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 ltcplt 16764

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-preset 16751 df-poset 16769 df-plt 16781

This theorem is referenced by: plttr 16793

W3C validator