Theorem psstrd 4084

Description: Proper subclass inclusion is transitive. Deduction form of psstr 4081. (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 4081	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 586	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊊ wpss 3937

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-in 3943  df-ss 3952  df-pss 3954

This theorem is referenced by: (None)