Theorem sspsstrd 3047

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3044. (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 3044	. 2 ⊢ ((A ⊆ B ∧ B ⊊ 𝐶) → A ⊊ 𝐶)
4	1, 2, 3	syl2anc 391	1 ⊢ (φ → A ⊊ 𝐶)

Syntax hints:   → wi 4   ⊆ wss 2911   ⊊ wpss 2912

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-11 1394  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-ne 2203  df-in 2918  df-ss 2925  df-pss 2927

This theorem is referenced by: (None)