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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 4696 | . . . 4 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
5 | 4 | simpld 112 | . . 3 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ 𝑆) |
6 | sonr 4335 | . . 3 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) | |
7 | 2, 5, 6 | sylancr 414 | . 2 ⊢ (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴) |
8 | 1, 7 | pm2.65i 640 | 1 ⊢ ¬ 𝐴𝑅𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 2160 ⊆ wss 3144 class class class wbr 4018 Or wor 4313 × cxp 4642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-14 2163 ax-ext 2171 ax-sep 4136 ax-pow 4192 ax-pr 4227 |
This theorem depends on definitions: df-bi 117 df-3an 982 df-tru 1367 df-nf 1472 df-sb 1774 df-clab 2176 df-cleq 2182 df-clel 2185 df-nfc 2321 df-ral 2473 df-rex 2474 df-v 2754 df-un 3148 df-in 3150 df-ss 3157 df-pw 3592 df-sn 3613 df-pr 3614 df-op 3616 df-br 4019 df-opab 4080 df-po 4314 df-iso 4315 df-xp 4650 |
This theorem is referenced by: son2lpi 5043 ltsonq 7427 genpdisj 7552 ltposr 7792 axpre-ltirr 7911 |
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