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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrntr | Structured version Visualization version GIF version |
Description: The covers relation is not transitive. (cvntr 30072 analog.) (Contributed by NM, 18-Jun-2012.) |
Ref | Expression |
---|---|
cvrntr.b | ⊢ 𝐵 = (Base‘𝐾) |
cvrntr.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
Ref | Expression |
---|---|
cvrntr | ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋𝐶𝑌 ∧ 𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrntr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2824 | . . . . 5 ⊢ (lt‘𝐾) = (lt‘𝐾) | |
3 | cvrntr.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | 1, 2, 3 | cvrlt 36410 | . . . 4 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → 𝑋(lt‘𝐾)𝑌) |
5 | 4 | ex 415 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 → 𝑋(lt‘𝐾)𝑌)) |
6 | 5 | 3adant3r3 1180 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → 𝑋(lt‘𝐾)𝑌)) |
7 | 1, 2, 3 | ltcvrntr 36564 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋(lt‘𝐾)𝑌 ∧ 𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍)) |
8 | 6, 7 | syland 604 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋𝐶𝑌 ∧ 𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 398 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 class class class wbr 5069 ‘cfv 6358 Basecbs 16486 ltcplt 17554 ⋖ ccvr 36402 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-mpt 5150 df-id 5463 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-iota 6317 df-fun 6360 df-fv 6366 df-covers 36406 |
This theorem is referenced by: atcvr0eq 36566 lnnat 36567 |
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