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| Mirrors > Home > MPE Home > Th. List > Mathboxes > ltcvrntr | Structured version Visualization version GIF version | ||
| Description: Non-transitive condition for the covers relation. (Contributed by NM, 18-Jun-2012.) | 
| Ref | Expression | 
|---|---|
| ltltncvr.b | ⊢ 𝐵 = (Base‘𝐾) | 
| ltltncvr.s | ⊢ < = (lt‘𝐾) | 
| ltltncvr.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) | 
| Ref | Expression | 
|---|---|
| ltcvrntr | ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ltltncvr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | ltltncvr.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
| 3 | ltltncvr.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
| 4 | 1, 2, 3 | cvrlt 39271 | . . . 4 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑌𝐶𝑍) → 𝑌 < 𝑍) | 
| 5 | 4 | ex 412 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌𝐶𝑍 → 𝑌 < 𝑍)) | 
| 6 | 5 | 3adant3r1 1183 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌𝐶𝑍 → 𝑌 < 𝑍)) | 
| 7 | 1, 2, 3 | ltltncvr 39425 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → ¬ 𝑋𝐶𝑍)) | 
| 8 | 6, 7 | sylan2d 605 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌𝐶𝑍) → ¬ 𝑋𝐶𝑍)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 ltcplt 18354 ⋖ ccvr 39263 | 
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-covers 39267 | 
| This theorem is referenced by: cvrntr 39427 | 
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