Theorem psssstrd 4066

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4063. (Contributed by David Moews, 1-May-2017.)

Hypotheses

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

Assertion

Ref	Expression
psssstrd	⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Proof of Theorem psssstrd

Step	Hyp	Ref	Expression
1		psssstrd.1	. 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵)
2		psssstrd.2	. 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶)
3		psssstr 4063	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 595	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3904   ⊊ wpss 3905

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-ex 1809  df-cleq 2754  df-ne 2958  df-ss 3921  df-pss 3924

This theorem is used by:  ackbij1lem15  10223  lsatssn0  39804  lsatexch  39845  lsatcvatlem  39851  lkrpssN  39965