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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 38009 | . . . 4 ⊢ (((𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑌𝐶𝑍) → 𝑌 < 𝑍) |
5 | 4 | ex 413 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌𝐶𝑍 → 𝑌 < 𝑍)) |
6 | 5 | 3adant3r1 1182 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌𝐶𝑍 → 𝑌 < 𝑍)) |
7 | 1, 2, 3 | ltltncvr 38163 | . 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 396 ∧ w3a 1087 = wceq 1541 ∈ wcel 2106 class class class wbr 5142 ‘cfv 6533 Basecbs 17128 ltcplt 18245 ⋖ ccvr 38001 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pow 5357 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5568 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-iota 6485 df-fun 6535 df-fv 6541 df-covers 38005 |
This theorem is referenced by: cvrntr 38165 |
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