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Mirrors > Home > MPE Home > Th. List > plelttr | Structured version Visualization version GIF version |
Description: Transitive law for chained "less than or equal to" and "less than". (sspsstr 4084 analog.) (Contributed by NM, 2-May-2012.) |
Ref | Expression |
---|---|
pltletr.b | ⊢ 𝐵 = (Base‘𝐾) |
pltletr.l | ⊢ ≤ = (le‘𝐾) |
pltletr.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
plelttr | ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 ≤ 𝑌 ∧ 𝑌 < 𝑍) → 𝑋 < 𝑍)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltletr.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | pltletr.l | . . . . 5 ⊢ ≤ = (le‘𝐾) | |
3 | pltletr.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
4 | 1, 2, 3 | pleval2 17577 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
5 | 4 | 3adant3r3 1180 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 ↔ (𝑋 < 𝑌 ∨ 𝑋 = 𝑌))) |
6 | 1, 3 | plttr 17582 | . . . . 5 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∧ 𝑌 < 𝑍) → 𝑋 < 𝑍)) |
7 | 6 | expd 418 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 < 𝑌 → (𝑌 < 𝑍 → 𝑋 < 𝑍))) |
8 | breq1 5071 | . . . . . 6 ⊢ (𝑋 = 𝑌 → (𝑋 < 𝑍 ↔ 𝑌 < 𝑍)) | |
9 | 8 | biimprd 250 | . . . . 5 ⊢ (𝑋 = 𝑌 → (𝑌 < 𝑍 → 𝑋 < 𝑍)) |
10 | 9 | a1i 11 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 = 𝑌 → (𝑌 < 𝑍 → 𝑋 < 𝑍))) |
11 | 7, 10 | jaod 855 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → ((𝑋 < 𝑌 ∨ 𝑋 = 𝑌) → (𝑌 < 𝑍 → 𝑋 < 𝑍))) |
12 | 5, 11 | sylbid 242 | . 2 ⊢ ((𝐾 ∈ Poset ∧ (𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵 ∧ 𝑍 ∈ 𝐵)) → (𝑋 ≤ 𝑌 → (𝑌 < 𝑍 → 𝑋 < 𝑍))) |
13 | 12 | impd 413 | 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 208 ∧ wa 398 ∨ wo 843 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 class class class wbr 5068 ‘cfv 6357 Basecbs 16485 lecple 16574 Posetcpo 17552 ltcplt 17553 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-proset 17540 df-poset 17558 df-plt 17570 |
This theorem is referenced by: isarchi3 30818 archiabllem2c 30826 athgt 36594 1cvratex 36611 |
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