Theorem psstrd 4039

Description: Proper subclass inclusion is transitive. Deduction form of psstr 4036. (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 4036	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 587	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊊ wpss 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-v 3425  df-in 3891  df-ss 3901  df-pss 3903

This theorem is referenced by: (None)