pltnle - Metamath Proof Explorer

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

Theorem pltnle 18056

Description: "Less than" implies not converse "less than or equal to". (Contributed by NM, 18-Oct-2011.)

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

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

**Proof of Theorem pltnle**
Step	Hyp	Ref	Expression
1		pleval2.l	. . . 4 ⊢ ≤ = (le‘𝐾)
2		pleval2.s	. . . 4 ⊢ < = (lt‘𝐾)
3	1, 2	pltval 18050	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌)))
4		pleval2.b	. . . . . . . 8 ⊢ 𝐵 = (Base‘𝐾)
5	4, 1	posasymb 18037	. . . . . . 7 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) ↔ 𝑋 = 𝑌))
6	5	biimpd 228	. . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑌 ≤ 𝑋) → 𝑋 = 𝑌))
7	6	expdimp 453	. . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑌 ≤ 𝑋 → 𝑋 = 𝑌))
8	7	necon3ad 2956	. . . 4 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 ≤ 𝑌) → (𝑋 ≠ 𝑌 → ¬ 𝑌 ≤ 𝑋))
9	8	expimpd 454	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ 𝑋 ≠ 𝑌) → ¬ 𝑌 ≤ 𝑋))
10	3, 9	sylbid 239	. 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 ≤ 𝑋))
11	10	imp 407	1 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌 ≤ 𝑋)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 Basecbs 16912 lecple 16969 Posetcpo 18025 ltcplt 18026

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-proset 18013 df-poset 18031 df-plt 18048

This theorem is referenced by: pltnlt 18058 pltn2lp 18059 ncvr1 37286 cvrnle 37294 atlrelat1 37335 cvrat 37436

W3C validator