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Mirrors > Home > MPE Home > Th. List > soirri | Structured version Visualization version 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 | soi.1 | . . . 4 ⊢ 𝑅 Or 𝑆 | |
2 | sonr 5491 | . . . 4 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) | |
3 | 1, 2 | mpan 688 | . . 3 ⊢ (𝐴 ∈ 𝑆 → ¬ 𝐴𝑅𝐴) |
4 | 3 | adantl 484 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) |
5 | soi.2 | . . . 4 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
6 | 5 | brel 5612 | . . 3 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | con3i 157 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) |
8 | 4, 7 | pm2.61i 184 | 1 ⊢ ¬ 𝐴𝑅𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 398 ∈ wcel 2110 ⊆ wss 3936 class class class wbr 5059 Or wor 5468 × cxp 5548 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pr 5322 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4562 df-pr 4564 df-op 4568 df-br 5060 df-opab 5122 df-po 5469 df-so 5470 df-xp 5556 |
This theorem is referenced by: son2lpi 5983 nqpr 10430 ltapr 10461 |
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