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| Mirrors > Home > MPE Home > Th. List > pltletr | Structured version Visualization version GIF version | ||
| Description: Transitive law for chained "less than" and "less than or equal to". (psssstr 4109 analog.) (Contributed by NM, 2-Dec-2011.) |
| Ref | Expression |
|---|---|
| pltletr.b | ⊢ 𝐵 = (Base‘𝐾) |
| pltletr.l | ⊢ ≤ = (le‘𝐾) |
| pltletr.s | ⊢ < = (lt‘𝐾) |
| Ref | Expression |
|---|---|
| pltletr | ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 < 𝑍)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | pltletr.b | . . . . . 6 ⊢ 𝐵 = (Base‘𝐾) | |
| 2 | pltletr.l | . . . . . 6 ⊢ ≤ = (le‘𝐾) | |
| 3 | pltletr.s | . . . . . 6 ⊢ < = (lt‘𝐾) | |
| 4 | 1, 2, 3 | pleval2 18382 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵) → (𝑌 ≤ 𝑍 ↔ (𝑌 < 𝑍 ∨ 𝑌 = 𝑍))) |
| 5 | 4 | 3adant3r1 1183 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑌 ≤ 𝑍 ↔ (𝑌 < 𝑍 ∨ 𝑌 = 𝑍))) |
| 6 | 5 | adantr 480 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 ≤ 𝑍 ↔ (𝑌 < 𝑍 ∨ 𝑌 = 𝑍))) |
| 7 | 1, 3 | plttr 18387 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → 𝑋 < 𝑍)) |
| 8 | 7 | expdimp 452 | . . . 4 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 < 𝑍 → 𝑋 < 𝑍)) |
| 9 | breq2 5147 | . . . . . 6 ⊢ (𝑌 = 𝑍 → (𝑋 < 𝑌 ↔ 𝑋 < 𝑍)) | |
| 10 | 9 | biimpcd 249 | . . . . 5 ⊢ (𝑋 < 𝑌 → (𝑌 = 𝑍 → 𝑋 < 𝑍)) |
| 11 | 10 | adantl 481 | . . . 4 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 = 𝑍 → 𝑋 < 𝑍)) |
| 12 | 8, 11 | jaod 860 | . . 3 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → ((𝑌 < 𝑍 ∨ 𝑌 = 𝑍) → 𝑋 < 𝑍)) |
| 13 | 6, 12 | sylbid 240 | . 2 ⊢ (((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) ∧ 𝑋 < 𝑌) → (𝑌 ≤ 𝑍 → 𝑋 < 𝑍)) |
| 14 | 13 | expimpd 453 | 1 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 ≤ 𝑍) → 𝑋 < 𝑍)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 848 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 class class class wbr 5143 ‘cfv 6561 Basecbs 17247 lecple 17304 Posetcpo 18353 ltcplt 18354 |
| 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-pr 5432 |
| 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-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 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-proset 18340 df-poset 18359 df-plt 18375 |
| This theorem is referenced by: cvrletrN 39274 atlen0 39311 atlelt 39440 2atlt 39441 ps-2 39480 llnnleat 39515 lplnnle2at 39543 lvolnle3at 39584 dalemcea 39662 2atm2atN 39787 dia2dimlem2 41067 dia2dimlem3 41068 |
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