Theorem sspsstrd 4109

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4106. (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 4106	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3949   ⊊ wpss 3950

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-v 3475  df-in 3956  df-ss 3966  df-pss 3968

This theorem is referenced by:  marypha1lem  9431  ackbij1lem15  10232  fin23lem38  10347  ltexprlem2  11035  mrieqv2d  17588