Theorem psssstrd 4060

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4057. (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 4057	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3898   ⊊ wpss 3899

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-cleq 2752  df-ne 2956  df-ss 3915  df-pss 3918

This theorem is used by:  ackbij1lem15  10282  lsatssn0  39979  lsatexch  40020  lsatcvatlem  40026  lkrpssN  40140