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Mirrors > Home > MPE Home > Th. List > Mathboxes > cvrnbtwn | Structured version Visualization version GIF version |
Description: There is no element between the two arguments of the covers relation. (cvnbtwn 28529 analog.) (Contributed by NM, 18-Oct-2011.) |
Ref | Expression |
---|---|
cvrfval.b | ⊢ 𝐵 = (Base‘𝐾) |
cvrfval.s | ⊢ < = (lt‘𝐾) |
cvrfval.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
Ref | Expression |
---|---|
cvrnbtwn | ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cvrfval.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | cvrfval.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
3 | cvrfval.c | . . . . 5 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | 1, 2, 3 | cvrval 33574 | . . . 4 ⊢ ((𝐾 ∈ 𝐴 ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)))) |
5 | 4 | 3adant3r3 1268 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 ↔ (𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)))) |
6 | ralnex 2975 | . . . . . . 7 ⊢ (∀𝑧 ∈ 𝐵 ¬ (𝑋 < 𝑧 ∧ 𝑧 < 𝑌) ↔ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)) | |
7 | breq2 4587 | . . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑋 < 𝑧 ↔ 𝑋 < 𝑍)) | |
8 | breq1 4586 | . . . . . . . . . 10 ⊢ (𝑧 = 𝑍 → (𝑧 < 𝑌 ↔ 𝑍 < 𝑌)) | |
9 | 7, 8 | anbi12d 743 | . . . . . . . . 9 ⊢ (𝑧 = 𝑍 → ((𝑋 < 𝑧 ∧ 𝑧 < 𝑌) ↔ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
10 | 9 | notbid 307 | . . . . . . . 8 ⊢ (𝑧 = 𝑍 → (¬ (𝑋 < 𝑧 ∧ 𝑧 < 𝑌) ↔ ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
11 | 10 | rspcv 3278 | . . . . . . 7 ⊢ (𝑍 ∈ 𝐵 → (∀𝑧 ∈ 𝐵 ¬ (𝑋 < 𝑧 ∧ 𝑧 < 𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
12 | 6, 11 | syl5bir 232 | . . . . . 6 ⊢ (𝑍 ∈ 𝐵 → (¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
13 | 12 | adantld 482 | . . . . 5 ⊢ (𝑍 ∈ 𝐵 → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
14 | 13 | 3ad2ant3 1077 | . . . 4 ⊢ ((𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
15 | 14 | adantl 481 | . . 3 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ ¬ ∃𝑧 ∈ 𝐵 (𝑋 < 𝑧 ∧ 𝑧 < 𝑌)) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
16 | 5, 15 | sylbid 229 | . 2 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋𝐶𝑌 → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌))) |
17 | 16 | 3impia 1253 | 1 ⊢ ((𝐾 ∈ 𝐴 ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) ∧ 𝑋𝐶𝑌) → ¬ (𝑋 < 𝑍 ∧ 𝑍 < 𝑌)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ∃wrex 2897 class class class wbr 4583 ‘cfv 5804 Basecbs 15695 ltcplt 16764 ⋖ ccvr 33567 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-iota 5768 df-fun 5806 df-fv 5812 df-covers 33571 |
This theorem is referenced by: cvrnbtwn2 33580 cvrnbtwn3 33581 cvrnbtwn4 33584 ltltncvr 33727 |
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