Theorem psssstrd 4044

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4041. (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 4041	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3887   ⊊ wpss 3888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ne 2944  df-v 3434  df-in 3894  df-ss 3904  df-pss 3906

This theorem is referenced by:  ackbij1lem15  9990  lsatssn0  37016  lsatexch  37057  lsatcvatlem  37063  lkrpssN  37177