Theorem psstrd 4090

Description: Proper subclass inclusion is transitive. Deduction form of psstr 4087. (Contributed by David Moews, 1-May-2017.)

Hypotheses

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

Assertion

Ref	Expression
psstrd	⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Proof of Theorem psstrd

Step	Hyp	Ref	Expression
1		psstrd.1	. 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵)
2		psstrd.2	. 2 ⊢ (𝜑 → 𝐵 ⊊ 𝐶)
3		psstr 4087	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊊ wpss 3932

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1779  df-cleq 2726  df-ne 2932  df-ss 3948  df-pss 3951

This theorem is referenced by: (None)