Theorem psssstrd 4104

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4101. (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 4101	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3943   ⊊ wpss 3944

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-v 3470  df-in 3950  df-ss 3960  df-pss 3962

This theorem is referenced by:  ackbij1lem15  10231  lsatssn0  38385  lsatexch  38426  lsatcvatlem  38432  lkrpssN  38546