tltnle - Mathbox for Thierry Arnoux

	Mathbox for Thierry Arnoux		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > tltnle		Structured version Visualization version GIF version

Theorem tltnle 30651

Description: In a Toset, less-than is equivalent to not inverse "less than or equal to" see pltnle 17578. (Contributed by Thierry Arnoux, 11-Feb-2018.)

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

**Assertion**
Ref	Expression
tltnle	⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))

**Proof of Theorem tltnle**
Step	Hyp	Ref	Expression
1		tospos 30647	. . 3 ⊢ (𝐾 ∈ Toset → 𝐾 ∈ Poset)
2		tleile.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3		tleile.l	. . . 4 ⊢ ≤ = (le‘𝐾)
4		tltnle.s	. . . 4 ⊢ < = (lt‘𝐾)
5	2, 3, 4	pltval3 17579	. . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
6	1, 5	syl3an1 1159	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋)))
7	2, 3	tleile 30650	. . 3 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋))
8		ibar 531	. . . 4 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → (¬ 𝑌 ≤ 𝑋 ↔ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋)))
9		pm5.61 997	. . . 4 ⊢ (((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) ∧ ¬ 𝑌 ≤ 𝑋) ↔ (𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋))
10	8, 9	syl6rbb 290	. . 3 ⊢ ((𝑋 ≤ 𝑌 ∨ 𝑌 ≤ 𝑋) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋))
11	7, 10	syl 17	. 2 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ((𝑋 ≤ 𝑌 ∧ ¬ 𝑌 ≤ 𝑋) ↔ ¬ 𝑌 ≤ 𝑋))
12	6, 11	bitrd 281	1 ⊢ ((𝐾 ∈ Toset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 ↔ ¬ 𝑌 ≤ 𝑋))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 Posetcpo 17552 ltcplt 17553 Tosetctos 17645

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332

This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-proset 17540 df-poset 17558 df-plt 17570 df-toset 17646

This theorem is referenced by: tlt2 30653 toslublem 30656 tosglblem 30658 isarchi2 30816 archirng 30819 archiabllem2c 30826 archiabl 30829

W3C validator