Theorem sspsstrd 3378

Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3375. (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 3375	. 2 ⊢ ((A ⊆ B ∧ B ⊊ C) → A ⊊ C)
4	1, 2, 3	syl2anc 642	1 ⊢ (φ → A ⊊ C)

Syntax hints:   → wi 4   ⊆ wss 3258   ⊊ wpss 3259

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pss 3262

This theorem is referenced by: (None)