Theorem sstr2 3280

Description: Transitivity of subclasses. Exercise 5 of [TakeutiZaring] p. 17. (Contributed by NM, 5-Aug-1993.) (Proof shortened by Andrew Salmon, 14-Jun-2011.)

Assertion

Ref	Expression
sstr2	⊢ (A ⊆ B → (B ⊆ C → A ⊆ C))

Proof of Theorem sstr2

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3268	. . . 4 ⊢ (A ⊆ B → (x ∈ A → x ∈ B))
2	1	imim1d 69	. . 3 ⊢ (A ⊆ B → ((x ∈ B → x ∈ C) → (x ∈ A → x ∈ C)))
3	2	alimdv 1621	. 2 ⊢ (A ⊆ B → (∀x(x ∈ B → x ∈ C) → ∀x(x ∈ A → x ∈ C)))
4		dfss2 3263	. 2 ⊢ (B ⊆ C ↔ ∀x(x ∈ B → x ∈ C))
5		dfss2 3263	. 2 ⊢ (A ⊆ C ↔ ∀x(x ∈ A → x ∈ C))
6	3, 4, 5	3imtr4g 261	1 ⊢ (A ⊆ B → (B ⊆ C → A ⊆ C))

Syntax hints:   → wi 4  ∀wal 1540   ∈ wcel 1710   ⊆ wss 3258

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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by:  sstr  3281  sstri  3282  sseq1  3293  sseq2  3294  ssun3  3429  ssun4  3430  ssinss1  3484  ssdisj  3601  sspwb  4119  funss  5127  funimass2  5171  fss  5231