Mathbox for Norm Megill |
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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 2610 | . . . 4 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | ncvr1.u | . . . 4 ⊢ 1 = (1.‘𝐾) | |
4 | 1, 2, 3 | ople1 33496 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → 𝑋(le‘𝐾) 1 ) |
5 | opposet 33486 | . . . . . 6 ⊢ (𝐾 ∈ OP → 𝐾 ∈ Poset) | |
6 | 5 | ad2antrr 758 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 𝐾 ∈ Poset) |
7 | 1, 3 | op1cl 33490 | . . . . . 6 ⊢ (𝐾 ∈ OP → 1 ∈ 𝐵) |
8 | 7 | ad2antrr 758 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 1 ∈ 𝐵) |
9 | simplr 788 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 𝑋 ∈ 𝐵) | |
10 | simpr 476 | . . . . 5 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → 1 (lt‘𝐾)𝑋) | |
11 | eqid 2610 | . . . . . 6 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
12 | 1, 2, 11 | pltnle 16789 | . . . . 5 ⊢ (((𝐾 ∈ Poset ∧ 1 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → ¬ 𝑋(le‘𝐾) 1 ) |
13 | 6, 8, 9, 10, 12 | syl31anc 1321 | . . . 4 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 (lt‘𝐾)𝑋) → ¬ 𝑋(le‘𝐾) 1 ) |
14 | 13 | ex 449 | . . 3 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ( 1 (lt‘𝐾)𝑋 → ¬ 𝑋(le‘𝐾) 1 )) |
15 | 4, 14 | mt2d 130 | . 2 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 (lt‘𝐾)𝑋) |
16 | simpll 786 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 𝐾 ∈ OP) | |
17 | 7 | ad2antrr 758 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 ∈ 𝐵) |
18 | simplr 788 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 𝑋 ∈ 𝐵) | |
19 | simpr 476 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 𝐶𝑋) | |
20 | ncvr1.c | . . . 4 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
21 | 1, 11, 20 | cvrlt 33575 | . . 3 ⊢ (((𝐾 ∈ OP ∧ 1 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 (lt‘𝐾)𝑋) |
22 | 16, 17, 18, 19, 21 | syl31anc 1321 | . 2 ⊢ (((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) ∧ 1 𝐶𝑋) → 1 (lt‘𝐾)𝑋) |
23 | 15, 22 | mtand 689 | 1 ⊢ ((𝐾 ∈ OP ∧ 𝑋 ∈ 𝐵) → ¬ 1 𝐶𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 lecple 15775 Posetcpo 16763 ltcplt 16764 1.cp1 16861 OPcops 33477 ⋖ ccvr 33567 |
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-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
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-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 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-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-preset 16751 df-poset 16769 df-plt 16781 df-lub 16797 df-p1 16863 df-oposet 33481 df-covers 33571 |
This theorem is referenced by: lhp2lt 34305 |
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