Metamath Proof Explorer

Theorem ssnpss

Description: Partial trichotomy law for subclasses. (Contributed by NM, 16-May-1996) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Assertion ssnpss ( 𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴 )

Proof

Step Hyp Ref Expression
1 dfpss3 ⊢ ( 𝐵 ⊊ 𝐴 ↔ ( 𝐵 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐵 ) )
2 1 simprbi ⊢ ( 𝐵 ⊊ 𝐴 → ¬ 𝐴 ⊆ 𝐵 )
3 2 con2i ⊢ ( 𝐴 ⊆ 𝐵 → ¬ 𝐵 ⊊ 𝐴 )