Theorem sspsstrd 4039

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4036. (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 4036	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 583	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3883   ⊊ wpss 3884

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-in 3890  df-ss 3900  df-pss 3902

This theorem is referenced by:  marypha1lem  9122  ackbij1lem15  9921  fin23lem38  10036  ltexprlem2  10724  mrieqv2d  17265