Theorem intun 1989

Description: The class intersection of the union of two classes. Theorem 78 of [Suppes] p. 42.

Assertion

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

Proof of Theorem intun

Step	Hyp	Ref	Expression
1		19.26 749	. . . 4 ⊢ (∀y((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)) ↔ (∀y(y ∈ A → x ∈ y) ∧ ∀y(y ∈ B → x ∈ y)))
2		elun 1601	. . . . . . 7 ⊢ (y ∈ (A ∪ B) ↔ (y ∈ A ∨ y ∈ B))
3	2	imbi1i 161	. . . . . 6 ⊢ ((y ∈ (A ∪ B) → x ∈ y) ↔ ((y ∈ A ∨ y ∈ B) → x ∈ y))
4		jaob 328	. . . . . 6 ⊢ (((y ∈ A ∨ y ∈ B) → x ∈ y) ↔ ((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)))
5	3, 4	bitr 151	. . . . 5 ⊢ ((y ∈ (A ∪ B) → x ∈ y) ↔ ((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)))
6	5	bial 695	. . . 4 ⊢ (∀y(y ∈ (A ∪ B) → x ∈ y) ↔ ∀y((y ∈ A → x ∈ y) ∧ (y ∈ B → x ∈ y)))
7		visset 1350	. . . . . 6 ⊢ x ∈ V
8	7	elint 1971	. . . . 5 ⊢ (x ∈ ∩A ↔ ∀y(y ∈ A → x ∈ y))
9	7	elint 1971	. . . . 5 ⊢ (x ∈ ∩B ↔ ∀y(y ∈ B → x ∈ y))
10	8, 9	anbi12i 369	. . . 4 ⊢ ((x ∈ ∩A ∧ x ∈ ∩B) ↔ (∀y(y ∈ A → x ∈ y) ∧ ∀y(y ∈ B → x ∈ y)))
11	1, 6, 10	3bitr4 158	. . 3 ⊢ (∀y(y ∈ (A ∪ B) → x ∈ y) ↔ (x ∈ ∩A ∧ x ∈ ∩B))
12	7	elint 1971	. . 3 ⊢ (x ∈ ∩(A ∪ B) ↔ ∀y(y ∈ (A ∪ B) → x ∈ y))
13		elin 1635	. . 3 ⊢ (x ∈ (∩A ∩ ∩B) ↔ (x ∈ ∩A ∧ x ∈ ∩B))
14	11, 12, 13	3bitr4 158	. 2 ⊢ (x ∈ ∩(A ∪ B) ↔ x ∈ (∩A ∩ ∩B))
15	14	cleqri 1101	1 ⊢ ∩(A ∪ B) = (∩A ∩ ∩B)

Syntax hints:   → wi 2   ∨ wo 195   ∧ wa 196  ∀wal 672   ∈ wel 803   = wceq 1091   ∈ wcel 1092   ∪ cun 1485   ∩ cin 1486  ∩cint 1965

This theorem is referenced by:  intunsn 1993

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-in 1491  df-int 1966

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