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Mirrors > Home > ILE Home > Th. List > sonr | GIF version |
Description: A strict order relation is irreflexive. (Contributed by NM, 24-Nov-1995.) |
Ref | Expression |
---|---|
sonr | ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sopo 4285 | . 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
2 | poirr 4279 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) | |
3 | 1, 2 | sylan 281 | 1 ⊢ ((𝑅 Or 𝐴 ∧ 𝐵 ∈ 𝐴) → ¬ 𝐵𝑅𝐵) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∈ wcel 2135 class class class wbr 3976 Po wpo 4266 Or wor 4267 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-ext 2146 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-v 2723 df-un 3115 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-po 4268 df-iso 4269 |
This theorem is referenced by: sotricim 4295 sotritrieq 4297 soirri 4992 addnqprlemfl 7491 addnqprlemfu 7492 mulnqprlemfl 7507 mulnqprlemfu 7508 1ne0sr 7698 |
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