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Mirrors > Home > MPE Home > Th. List > pltn2lp | Structured version Visualization version GIF version |
Description: The less-than relation has no 2-cycle loops. (pssn2lp 3968 analog.) (Contributed by NM, 2-Dec-2011.) |
Ref | Expression |
---|---|
pltnlt.b | ⊢ 𝐵 = (Base‘𝐾) |
pltnlt.s | ⊢ < = (lt‘𝐾) |
Ref | Expression |
---|---|
pltn2lp | ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pltnlt.b | . . . . 5 ⊢ 𝐵 = (Base‘𝐾) | |
2 | eqid 2778 | . . . . 5 ⊢ (le‘𝐾) = (le‘𝐾) | |
3 | pltnlt.s | . . . . 5 ⊢ < = (lt‘𝐾) | |
4 | 1, 2, 3 | pltnle 17434 | . . . 4 ⊢ (((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) ∧ 𝑋 < 𝑌) → ¬ 𝑌(le‘𝐾)𝑋) |
5 | 4 | ex 405 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌(le‘𝐾)𝑋)) |
6 | 2, 3 | pltle 17429 | . . . 4 ⊢ ((𝐾 ∈ Poset ∧ 𝑌 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) → (𝑌 < 𝑋 → 𝑌(le‘𝐾)𝑋)) |
7 | 6 | 3com23 1106 | . . 3 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑌 < 𝑋 → 𝑌(le‘𝐾)𝑋)) |
8 | 5, 7 | nsyld 156 | . 2 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → (𝑋 < 𝑌 → ¬ 𝑌 < 𝑋)) |
9 | imnan 391 | . 2 ⊢ ((𝑋 < 𝑌 → ¬ 𝑌 < 𝑋) ↔ ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) | |
10 | 8, 9 | sylib 210 | 1 ⊢ ((𝐾 ∈ Poset ∧ 𝑋 ∈ 𝐵 ∧ 𝑌 ∈ 𝐵) → ¬ (𝑋 < 𝑌 ∧ 𝑌 < 𝑋)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 387 ∧ w3a 1068 = wceq 1507 ∈ wcel 2050 class class class wbr 4929 ‘cfv 6188 Basecbs 16339 lecple 16428 Posetcpo 17408 ltcplt 17409 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2750 ax-sep 5060 ax-nul 5067 ax-pr 5186 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-ne 2968 df-ral 3093 df-rex 3094 df-rab 3097 df-v 3417 df-sbc 3682 df-dif 3832 df-un 3834 df-in 3836 df-ss 3843 df-nul 4179 df-if 4351 df-sn 4442 df-pr 4444 df-op 4448 df-uni 4713 df-br 4930 df-opab 4992 df-mpt 5009 df-id 5312 df-xp 5413 df-rel 5414 df-cnv 5415 df-co 5416 df-dm 5417 df-iota 6152 df-fun 6190 df-fv 6196 df-proset 17396 df-poset 17414 df-plt 17426 |
This theorem is referenced by: plttr 17438 |
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