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Mirrors > Home > ILE Home > Th. List > soirri | GIF version |
Description: A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
soi.1 | ⊢ 𝑅 Or 𝑆 |
soi.2 | ⊢ 𝑅 ⊆ (𝑆 × 𝑆) |
Ref | Expression |
---|---|
soirri | ⊢ ¬ 𝐴𝑅𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | id 19 | . 2 ⊢ (𝐴𝑅𝐴 → 𝐴𝑅𝐴) | |
2 | soi.1 | . . 3 ⊢ 𝑅 Or 𝑆 | |
3 | soi.2 | . . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
4 | 3 | brel 4447 | . . . 4 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
5 | 4 | simpld 110 | . . 3 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ 𝑆) |
6 | sonr 4107 | . . 3 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) | |
7 | 2, 5, 6 | sylancr 405 | . 2 ⊢ (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴) |
8 | 1, 7 | pm2.65i 601 | 1 ⊢ ¬ 𝐴𝑅𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1434 ⊆ wss 2984 class class class wbr 3811 Or wor 4085 × cxp 4398 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2614 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-br 3812 df-opab 3866 df-po 4086 df-iso 4087 df-xp 4406 |
This theorem is referenced by: son2lpi 4782 ltsonq 6859 genpdisj 6984 ltposr 7211 axpre-ltirr 7319 |
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