Theorem psstrd 4107

Description: Proper subclass inclusion is transitive. Deduction form of psstr 4104. (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 4104	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊊ wpss 3949

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-v 3476  df-in 3955  df-ss 3965  df-pss 3967

This theorem is referenced by: (None)