Theorem sspsstrd 4084

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 4081. (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 4081	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶)
4	1, 2, 3	syl2anc 586	1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 3935   ⊊ wpss 3936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-ne 3017  df-in 3942  df-ss 3951  df-pss 3953

This theorem is referenced by:  marypha1lem  8891  ackbij1lem15  9650  fin23lem38  9765  ltexprlem2  10453  mrieqv2d  16904