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Mirrors > Home > MPE Home > Th. List > poirr | Structured version Visualization version GIF version |
Description: A partial order relation is irreflexive. (Contributed by NM, 27-Mar-1997.) |
Ref | Expression |
---|---|
poirr | ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-3an 1033 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) ↔ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵 ∈ 𝐴)) | |
2 | anabs1 846 | . . 3 ⊢ (((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) ∧ 𝐵 ∈ 𝐴) ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) | |
3 | anidm 674 | . . 3 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) ↔ 𝐵 ∈ 𝐴) | |
4 | 1, 2, 3 | 3bitrri 286 | . 2 ⊢ (𝐵 ∈ 𝐴 ↔ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) |
5 | pocl 4966 | . . . 4 ⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵 ∧ 𝐵𝑅𝐵) → 𝐵𝑅𝐵)))) | |
6 | 5 | imp 444 | . . 3 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐵 ∧ 𝐵𝑅𝐵) → 𝐵𝑅𝐵))) |
7 | 6 | simpld 474 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴 ∧ 𝐵 ∈ 𝐴)) → ¬ 𝐵𝑅𝐵) |
8 | 4, 7 | sylan2b 491 | 1 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 ∧ w3a 1031 ∈ wcel 1977 class class class wbr 4583 Po wpo 4957 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-po 4959 |
This theorem is referenced by: po2nr 4972 pofun 4975 sonr 4980 poirr2 5439 predpoirr 5625 soisoi 6478 poxp 7176 swoer 7659 frfi 8090 wemappo 8337 zorn2lem3 9203 ex-po 26684 pocnv 30907 poseq 30994 ipo0 37674 |
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