Theorem sspsstrd 4042

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4039. (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 4039	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 596	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

This proof depends on syntax axioms:   → wi 4   ⊆ wss 3900   ⊊ wpss 4016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-ex 1813  df-cleq 2755  df-ne 2960  df-ss 3901  df-pss 4017

This theorem is used by:  marypha1lem  9455  ackbij1lem15  10341  fin23lem38  10457  ltexprlem2  11152  mrieqv2d  17845