Metamath Proof Explorer


Theorem soirri

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

Proof

Step Hyp Ref Expression
1 soi.1 𝑅 Or 𝑆
2 soi.2 𝑅 ⊆ ( 𝑆 × 𝑆 )
3 sonr ( ( 𝑅 Or 𝑆𝐴𝑆 ) → ¬ 𝐴 𝑅 𝐴 )
4 1 3 mpan ( 𝐴𝑆 → ¬ 𝐴 𝑅 𝐴 )
5 4 adantl ( ( 𝐴𝑆𝐴𝑆 ) → ¬ 𝐴 𝑅 𝐴 )
6 2 brel ( 𝐴 𝑅 𝐴 → ( 𝐴𝑆𝐴𝑆 ) )
7 6 con3i ( ¬ ( 𝐴𝑆𝐴𝑆 ) → ¬ 𝐴 𝑅 𝐴 )
8 5 7 pm2.61i ¬ 𝐴 𝑅 𝐴