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Metamath Proof Explorer

Theorem sspsstrd

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr . (Contributed by David Moews, 1-May-2017)

Ref Expression
Hypotheses sspsstrd.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
sspsstrd.2 ⊢ ( 𝜑 → 𝐵 ⊊ 𝐶 )
Assertion sspsstrd ( 𝜑 → 𝐴 ⊊ 𝐶 )

Proof

Step Hyp Ref Expression
1 sspsstrd.1 ⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2 sspsstrd.2 ⊢ ( 𝜑 → 𝐵 ⊊ 𝐶 )
3 sspsstr ⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶 ) → 𝐴 ⊊ 𝐶 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → 𝐴 ⊊ 𝐶 )