Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
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 5489 | . . . 4 ⊢ ((𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) | |
3 | 1, 2 | mpan 686 | . . 3 ⊢ (𝐴 ∈ 𝑆 → ¬ 𝐴𝑅𝐴) |
4 | 3 | adantl 482 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) |
5 | soi.2 | . . . 4 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
6 | 5 | brel 5610 | . . 3 ⊢ (𝐴𝑅𝐴 → (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
7 | 6 | con3i 157 | . 2 ⊢ (¬ (𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → ¬ 𝐴𝑅𝐴) |
8 | 4, 7 | pm2.61i 183 | 1 ⊢ ¬ 𝐴𝑅𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 class class class wbr 5057 Or wor 5466 × cxp 5546 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-br 5058 df-opab 5120 df-po 5467 df-so 5468 df-xp 5554 |
This theorem is referenced by: son2lpi 5981 nqpr 10424 ltapr 10455 |
Copyright terms: Public domain | W3C validator |