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Metamath Proof Explorer

Theorem psstrd

Description: Proper subclass inclusion is transitive. Deduction form of psstr . (Contributed by David Moews, 1-May-2017)

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

Proof

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