Theorem sspsstrd 4074

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4071. (Contributed by David Moews, 1-May-2017.)

Hypotheses

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

Assertion

Ref	Expression
sspsstrd	⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Proof of Theorem sspsstrd

Step	Hyp	Ref	Expression
1		sspsstrd.1	. 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sspsstrd.2	. 2 ⊢ (𝜑 → 𝐵 ⊊ 𝐶)
3		sspsstr 4071	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 584	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3914   ⊊ wpss 3915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-ex 1780  df-cleq 2721  df-ne 2926  df-ss 3931  df-pss 3934

This theorem is referenced by:  marypha1lem  9384  ackbij1lem15  10186  fin23lem38  10302  ltexprlem2  10990  mrieqv2d  17600