Description: A strict order relation is irreflexive. (Contributed by NM, 10-Feb-1996) (Revised by Mario Carneiro, 10-May-2013)
Ref | Expression | ||
---|---|---|---|
Hypotheses | soi.1 | ⊢ 𝑅 Or 𝑆 | |
soi.2 | ⊢ 𝑅 ⊆ ( 𝑆 × 𝑆 ) | ||
Assertion | soirri | ⊢ ¬ 𝐴 𝑅 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | soi.1 | ⊢ 𝑅 Or 𝑆 | |
2 | soi.2 | ⊢ 𝑅 ⊆ ( 𝑆 × 𝑆 ) | |
3 | sonr | ⊢ ( ( 𝑅 Or 𝑆 ∧ 𝐴 ∈ 𝑆 ) → ¬ 𝐴 𝑅 𝐴 ) | |
4 | 1 3 | mpan | ⊢ ( 𝐴 ∈ 𝑆 → ¬ 𝐴 𝑅 𝐴 ) |
5 | 4 | adantl | ⊢ ( ( 𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) → ¬ 𝐴 𝑅 𝐴 ) |
6 | 2 | brel | ⊢ ( 𝐴 𝑅 𝐴 → ( 𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) ) |
7 | 6 | con3i | ⊢ ( ¬ ( 𝐴 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆 ) → ¬ 𝐴 𝑅 𝐴 ) |
8 | 5 7 | pm2.61i | ⊢ ¬ 𝐴 𝑅 𝐴 |