Theorem intun 3959

Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42. (Contributed by NM, 22-Sep-2002.)

Assertion

Ref	Expression
intun	⊢ ∩(A ∪ B) = (∩A ∩ ∩B)

Proof of Theorem intun

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.26 1593	. . . 4 ⊢ (∀y((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)) ↔ (∀y(y ∈ A → x ∈ y) ∧ ∀y(y ∈ B → x ∈ y)))
2		elun 3221	. . . . . . 7 ⊢ (y ∈ (A ∪ B) ↔ (y ∈ A ∨ y ∈ B))
3	2	imbi1i 315	. . . . . 6 ⊢ ((y ∈ (A ∪ B) → x ∈ y) ↔ ((y ∈ A ∨ y ∈ B) → x ∈ y))
4		jaob 758	. . . . . 6 ⊢ (((y ∈ A ∨ y ∈ B) → x ∈ y) ↔ ((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)))
5	3, 4	bitri 240	. . . . 5 ⊢ ((y ∈ (A ∪ B) → x ∈ y) ↔ ((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)))
6	5	albii 1566	. . . 4 ⊢ (∀y(y ∈ (A ∪ B) → x ∈ y) ↔ ∀y((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)))
7		vex 2863	. . . . . 6 ⊢ x ∈ V
8	7	elint 3933	. . . . 5 ⊢ (x ∈ ∩A ↔ ∀y(y ∈ A → x ∈ y))
9	7	elint 3933	. . . . 5 ⊢ (x ∈ ∩B ↔ ∀y(y ∈ B → x ∈ y))
10	8, 9	anbi12i 678	. . . 4 ⊢ ((x ∈ ∩A ∧ x ∈ ∩B) ↔ (∀y(y ∈ A → x ∈ y) ∧ ∀y(y ∈ B → x ∈ y)))
11	1, 6, 10	3bitr4i 268	. . 3 ⊢ (∀y(y ∈ (A ∪ B) → x ∈ y) ↔ (x ∈ ∩A ∧ x ∈ ∩B))
12	7	elint 3933	. . 3 ⊢ (x ∈ ∩(A ∪ B) ↔ ∀y(y ∈ (A ∪ B) → x ∈ y))
13		elin 3220	. . 3 ⊢ (x ∈ (∩A ∩ ∩B) ↔ (x ∈ ∩A ∧ x ∈ ∩B))
14	11, 12, 13	3bitr4i 268	. 2 ⊢ (x ∈ ∩(A ∪ B) ↔ x ∈ (∩A ∩ ∩B))
15	14	eqriv 2350	1 ⊢ ∩(A ∪ B) = (∩A ∩ ∩B)

Syntax hints:   → wi 4   ∨ wo 357   ∧ wa 358  ∀wal 1540   = wceq 1642   ∈ wcel 1710   ∪ cun 3208   ∩ cin 3209  ∩cint 3927

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-un 3215  df-int 3928

This theorem is referenced by:  intunsn  3966