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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 4490 | . . . 4 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
5 | 4 | simpld 110 | . . 3 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ 𝑆) |
6 | sonr 4144 | . . 3 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) | |
7 | 2, 5, 6 | sylancr 405 | . 2 ⊢ (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴) |
8 | 1, 7 | pm2.65i 603 | 1 ⊢ ¬ 𝐴𝑅𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1438 ⊆ wss 2999 class class class wbr 3845 Or wor 4122 × cxp 4436 |
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 579 ax-in2 580 ax-io 665 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-10 1441 ax-11 1442 ax-i12 1443 ax-bndl 1444 ax-4 1445 ax-14 1450 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 ax-sep 3957 ax-pow 4009 ax-pr 4036 |
This theorem depends on definitions: df-bi 115 df-3an 926 df-tru 1292 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-nfc 2217 df-ral 2364 df-rex 2365 df-v 2621 df-un 3003 df-in 3005 df-ss 3012 df-pw 3431 df-sn 3452 df-pr 3453 df-op 3455 df-br 3846 df-opab 3900 df-po 4123 df-iso 4124 df-xp 4444 |
This theorem is referenced by: son2lpi 4828 ltsonq 6957 genpdisj 7082 ltposr 7309 axpre-ltirr 7417 |
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