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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 5572 | . . . 4 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) | |
3 | 1, 2 | mpan 689 | . . 3 ⊢ (𝐴 ∈ 𝑆 → ¬ 𝐴𝑅𝐴) |
4 | 3 | adantl 483 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) |
5 | soi.2 | . . . 4 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
6 | 5 | brel 5701 | . . 3 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | con3i 154 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) |
8 | 4, 7 | pm2.61i 182 | 1 ⊢ ¬ 𝐴𝑅𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3914 class class class wbr 5109 Or wor 5548 × cxp 5635 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5260 ax-nul 5267 ax-pr 5388 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3449 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-sn 4591 df-pr 4593 df-op 4597 df-br 5110 df-opab 5172 df-po 5549 df-so 5550 df-xp 5643 |
This theorem is referenced by: son2lpi 6086 nqpr 10958 ltapr 10989 |
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