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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 2738 | . . 3 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
3 | cvrle.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | 1, 2, 3 | cvrlt 37048 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌) |
5 | cvrle.l | . . 3 ⊢ ≤ = (le‘𝐾) | |
6 | 1, 5, 2 | pltnle 17869 | . 2 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋(lt‘𝐾)𝑌) → ¬ 𝑌 ≤ 𝑋) |
7 | 4, 6 | syldan 594 | 1 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ 𝑌 ≤ 𝑋) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2111 class class class wbr 5068 ‘cfv 6398 Basecbs 16785 lecple 16834 Posetcpo 17839 ltcplt 17840 ⋖ ccvr 37040 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5207 ax-nul 5214 ax-pow 5273 ax-pr 5337 ax-un 7542 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3423 df-sbc 3710 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4253 df-if 4455 df-pw 4530 df-sn 4557 df-pr 4559 df-op 4563 df-uni 4835 df-br 5069 df-opab 5131 df-mpt 5151 df-id 5470 df-xp 5572 df-rel 5573 df-cnv 5574 df-co 5575 df-dm 5576 df-iota 6356 df-fun 6400 df-fv 6406 df-proset 17827 df-poset 17845 df-plt 17861 df-covers 37044 |
This theorem is referenced by: atnle0 37087 dih1 39064 |
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