Theorem psssstrd 3864

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 3861. (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 3861	. 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 565	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3721   ⊊ wpss 3722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-ext 2750

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-ne 2943  df-in 3728  df-ss 3735  df-pss 3737

This theorem is referenced by:  ackbij1lem15  9257  lsatssn0  34804  lsatexch  34845  lsatcvatlem  34851  lkrpssN  34965