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Mirrors > Home > MPE Home > Th. List > plttr | Structured version Visualization version GIF version |
Description: The less-than relation is transitive. (psstr 4039 analog.) (Contributed by NM, 2-Dec-2011.) |
Ref | Expression |
---|---|
pltnlt.b | ⊢ 𝐵 = (Base‘𝐾) |
pltnlt.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
plttr | ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → 𝑋 < 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2738 | . . . . . 6 ⊢ (le‘𝐾) = (le‘𝐾) | |
2 | pltnlt.s | . . . . . 6 ⊢ < = (lt‘𝐾) | |
3 | 1, 2 | pltle 18051 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → 𝑋(le‘𝐾)𝑌)) |
4 | 3 | 3adant3r3 1183 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 → 𝑋(le‘𝐾)𝑌)) |
5 | 1, 2 | pltle 18051 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 < 𝑍 → 𝑌(le‘𝐾)𝑍)) |
6 | 5 | 3adant3r1 1181 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 < 𝑍 → 𝑌(le‘𝐾)𝑍)) |
7 | pltnlt.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
8 | 7, 1 | postr 18038 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋(le‘𝐾)𝑌 ∧ 𝑌(le‘𝐾)𝑍) → 𝑋(le‘𝐾)𝑍)) |
9 | 4, 6, 8 | syl2and 608 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → 𝑋(le‘𝐾)𝑍)) |
10 | 7, 2 | pltn2lp 18059 | . . . . . 6 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) |
11 | 10 | 3adant3r3 1183 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) |
12 | breq2 5078 | . . . . . . 7 ⊢ (𝑋 = 𝑍 → (𝑌 < 𝑋 ↔ 𝑌 < 𝑍)) | |
13 | 12 | anbi2d 629 | . . . . . 6 ⊢ (𝑋 = 𝑍 → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑋) ↔ (𝑋 < 𝑌 ∧ 𝑌 < 𝑍))) |
14 | 13 | notbid 318 | . . . . 5 ⊢ (𝑋 = 𝑍 → (¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑍))) |
15 | 11, 14 | syl5ibcom 244 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 = 𝑍 → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑍))) |
16 | 15 | necon2ad 2958 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → 𝑋 ≠ 𝑍)) |
17 | 9, 16 | jcad 513 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → (𝑋(le‘𝐾)𝑍 ∧ 𝑋 ≠ 𝑍))) |
18 | 1, 2 | pltval 18050 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍 ∧ 𝑋 ≠ 𝑍))) |
19 | 18 | 3adant3r2 1182 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑍 ↔ (𝑋(le‘𝐾)𝑍 ∧ 𝑋 ≠ 𝑍))) |
20 | 17, 19 | sylibrd 258 | 1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → 𝑋 < 𝑍)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ≠ wne 2943 class class class wbr 5074 ‘cfv 6433 Basecbs 16912 lecple 16969 Posetcpo 18025 ltcplt 18026 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 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 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-iota 6391 df-fun 6435 df-fv 6441 df-proset 18013 df-poset 18031 df-plt 18048 |
This theorem is referenced by: pltletr 18061 plelttr 18062 pospo 18063 archiabllem2c 31449 ofldchr 31513 hlhgt2 37403 hl0lt1N 37404 lhp0lt 38017 |
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