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Mirrors > Home > MPE Home > Th. List > Mathboxes > ncvr1 | Structured version Visualization version GIF version |
Description: No element covers the lattice unit. (Contributed by NM, 8-Jul-2013.) |
Ref | Expression |
---|---|
ncvr1.b | ⊢ 𝐵 = (Base‘𝐾) |
ncvr1.u | ⊢ 1 = (1.‘𝐾) |
ncvr1.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
Ref | Expression |
---|---|
ncvr1 | ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 𝐶𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ncvr1.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2770 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | ncvr1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
4 | 1, 2, 3 | ople1 34993 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾) 1 ) |
5 | opposet 34983 | . . . . . 6 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
6 | 5 | ad2antrr 697 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 𝐾 ∈ Poset) |
7 | 1, 3 | op1cl 34987 | . . . . . 6 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
8 | 7 | ad2antrr 697 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 1 ∈ 𝐵) |
9 | simplr 744 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 𝑋 ∈ 𝐵) | |
10 | simpr 471 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 1 (lt‘𝐾)𝑋) | |
11 | eqid 2770 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
12 | 1, 2, 11 | pltnle 17173 | . . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 1 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → ¬ 𝑋(le‘𝐾) 1 ) |
13 | 6, 8, 9, 10, 12 | syl31anc 1478 | . . . 4 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → ¬ 𝑋(le‘𝐾) 1 ) |
14 | 13 | ex 397 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 (lt‘𝐾)𝑋 → ¬ 𝑋(le‘𝐾) 1 )) |
15 | 4, 14 | mt2d 133 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 (lt‘𝐾)𝑋) |
16 | simpll 742 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 𝐾 ∈ OP) | |
17 | 7 | ad2antrr 697 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 ∈ 𝐵) |
18 | simplr 744 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 𝑋 ∈ 𝐵) | |
19 | simpr 471 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 𝐶𝑋) | |
20 | ncvr1.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
21 | 1, 11, 20 | cvrlt 35072 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 1 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 (lt‘𝐾)𝑋) |
22 | 16, 17, 18, 19, 21 | syl31anc 1478 | . 2 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 (lt‘𝐾)𝑋) |
23 | 15, 22 | mtand 799 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 𝐶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 382 = wceq 1630 ∈ wcel 2144 class class class wbr 4784 ‘cfv 6031 Basecbs 16063 lecple 16155 Posetcpo 17147 ltcplt 17148 1.cp1 17245 OPcops 34974 ⋖ ccvr 35064 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-rep 4902 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-id 5157 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-riota 6753 df-ov 6795 df-preset 17135 df-poset 17153 df-plt 17165 df-lub 17181 df-p1 17247 df-oposet 34978 df-covers 35068 |
This theorem is referenced by: lhp2lt 35802 |
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