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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 4599 | . . . 4 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
5 | 4 | simpld 111 | . . 3 ⊢ (𝐴𝑅𝐴 → 𝐴 ∈ 𝑆) |
6 | sonr 4247 | . . 3 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) | |
7 | 2, 5, 6 | sylancr 411 | . 2 ⊢ (𝐴𝑅𝐴 → ¬ 𝐴𝑅𝐴) |
8 | 1, 7 | pm2.65i 629 | 1 ⊢ ¬ 𝐴𝑅𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∈ wcel 1481 ⊆ wss 3076 class class class wbr 3937 Or wor 4225 × cxp 4545 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-br 3938 df-opab 3998 df-po 4226 df-iso 4227 df-xp 4553 |
This theorem is referenced by: son2lpi 4943 ltsonq 7230 genpdisj 7355 ltposr 7595 axpre-ltirr 7714 |
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