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Mirrors > Home > MPE Home > Th. List > po2nr | Structured version Visualization version GIF version |
Description: A partial order has no 2-cycle loops. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
po2nr | ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | poirr 5600 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | |
2 | 1 | adantrr 715 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵) |
3 | potr 5601 | . . . . . 6 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)) | |
4 | 3 | 3exp2 1354 | . . . . 5 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → (𝐵 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))))) |
5 | 4 | com34 91 | . . . 4 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵))))) |
6 | 5 | pm2.43d 53 | . . 3 ⊢ (𝑅 Po 𝐴 → (𝐵 ∈ 𝐴 → (𝐶 ∈ 𝐴 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)))) |
7 | 6 | imp32 419 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵) → 𝐵𝑅𝐵)) |
8 | 2, 7 | mtod 197 | 1 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴)) → ¬ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ∈ wcel 2106 class class class wbr 5148 Po wpo 5586 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-po 5588 |
This theorem is referenced by: po3nr 5603 so2nr 5614 soisoi 7324 poxp2 8128 poxp3 8135 wemaplem2 9541 pospo 18297 poprelb 46182 |
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