Theorem ssin 3478

Description: Subclass of intersection. Theorem 2.8(vii) of [Monk1] p. 26. (Contributed by NM, 15-Jun-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.)

Assertion

Ref	Expression
ssin	⊢ ((A ⊆ B ∧ A ⊆ C) ↔ A ⊆ (B ∩ C))

Proof of Theorem ssin

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3220	. . . . 5 ⊢ (x ∈ (B ∩ C) ↔ (x ∈ B ∧ x ∈ C))
2	1	imbi2i 303	. . . 4 ⊢ ((x ∈ A → x ∈ (B ∩ C)) ↔ (x ∈ A → (x ∈ B ∧ x ∈ C)))
3	2	albii 1566	. . 3 ⊢ (∀x(x ∈ A → x ∈ (B ∩ C)) ↔ ∀x(x ∈ A → (x ∈ B ∧ x ∈ C)))
4		jcab 833	. . . 4 ⊢ ((x ∈ A → (x ∈ B ∧ x ∈ C)) ↔ ((x ∈ A → x ∈ B) ∧ (x ∈ A → x ∈ C)))
5	4	albii 1566	. . 3 ⊢ (∀x(x ∈ A → (x ∈ B ∧ x ∈ C)) ↔ ∀x((x ∈ A → x ∈ B) ∧ (x ∈ A → x ∈ C)))
6		19.26 1593	. . 3 ⊢ (∀x((x ∈ A → x ∈ B) ∧ (x ∈ A → x ∈ C)) ↔ (∀x(x ∈ A → x ∈ B) ∧ ∀x(x ∈ A → x ∈ C)))
7	3, 5, 6	3bitrri 263	. 2 ⊢ ((∀x(x ∈ A → x ∈ B) ∧ ∀x(x ∈ A → x ∈ C)) ↔ ∀x(x ∈ A → x ∈ (B ∩ C)))
8		dfss2 3263	. . 3 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B))
9		dfss2 3263	. . 3 ⊢ (A ⊆ C ↔ ∀x(x ∈ A → x ∈ C))
10	8, 9	anbi12i 678	. 2 ⊢ ((A ⊆ B ∧ A ⊆ C) ↔ (∀x(x ∈ A → x ∈ B) ∧ ∀x(x ∈ A → x ∈ C)))
11		dfss2 3263	. 2 ⊢ (A ⊆ (B ∩ C) ↔ ∀x(x ∈ A → x ∈ (B ∩ C)))
12	7, 10, 11	3bitr4i 268	1 ⊢ ((A ⊆ B ∧ A ⊆ C) ↔ A ⊆ (B ∩ C))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358  ∀wal 1540   ∈ wcel 1710   ∩ cin 3209   ⊆ 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:  ssini  3479  ssind  3480  uneqin  3507  disjpss  3602  fin  5247  clos1induct  5881  sbthlem1  6204