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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrnle | Structured version Visualization version GIF version |
Description: The covers relation implies the negation of the converse "less than or equal to" relation. (Contributed by NM, 18-Oct-2011.) |
Ref | Expression |
---|---|
cvrle.b | ⊢ 𝐵 = (Base‘𝐾) |
cvrle.l | ⊢ ≤ = (le‘𝐾) |
cvrle.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
Ref | Expression |
---|---|
cvrnle | ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 ≤ 𝑋) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrle.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2734 | . . 3 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
3 | cvrle.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | 1, 2, 3 | cvrlt 39251 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌) |
5 | cvrle.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | 1, 5, 2 | pltnle 18395 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋(lt‘𝐾)𝑌) → ¬ 𝑌 ≤ 𝑋) |
7 | 4, 6 | syldan 591 | 1 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1536 ∈ wcel 2105 class class class wbr 5147 ‘cfv 6562 Basecbs 17244 lecple 17304 Posetcpo 18364 ltcplt 18365 ⋖ ccvr 39243 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pow 5370 ax-pr 5437 ax-un 7753 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-sbc 3791 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-iota 6515 df-fun 6564 df-fv 6570 df-proset 18351 df-poset 18370 df-plt 18387 df-covers 39247 |
This theorem is referenced by: atnle0 39290 dih1 41268 |
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