Theorem iinun2 2031

Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 2027 to recover Enderton's theorem.

Assertion

Ref	Expression
iinun2	⊢ ∩x ∈ A (B ∪ C) = (B ∪ ∩x ∈ A C)

Distinct variable group(s):   x,A   x,B

Proof of Theorem iinun2

Step	Hyp	Ref	Expression
1		r19.32v 1297	. . . 4 ⊢ (∀x ∈ A (y ∈ B ∨ y ∈ C) ↔ (y ∈ B ∨ ∀x ∈ A y ∈ C))
2		elun 1601	. . . . 5 ⊢ (y ∈ (B ∪ C) ↔ (y ∈ B ∨ y ∈ C))
3	2	biral 1223	. . . 4 ⊢ (∀x ∈ A y ∈ (B ∪ C) ↔ ∀x ∈ A (y ∈ B ∨ y ∈ C))
4		visset 1350	. . . . . 6 ⊢ y ∈ V
5		eliin 1999	. . . . . 6 ⊢ (y ∈ V → (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C))
6	4, 5	ax-mp 6	. . . . 5 ⊢ (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C)
7	6	orbi2i 214	. . . 4 ⊢ ((y ∈ B ∨ y ∈ ∩x ∈ A C) ↔ (y ∈ B ∨ ∀x ∈ A y ∈ C))
8	1, 3, 7	3bitr4 158	. . 3 ⊢ (∀x ∈ A y ∈ (B ∪ C) ↔ (y ∈ B ∨ y ∈ ∩x ∈ A C))
9		eliin 1999	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A (B ∪ C) ↔ ∀x ∈ A y ∈ (B ∪ C)))
10	4, 9	ax-mp 6	. . 3 ⊢ (y ∈ ∩x ∈ A (B ∪ C) ↔ ∀x ∈ A y ∈ (B ∪ C))
11		elun 1601	. . 3 ⊢ (y ∈ (B ∪ ∩x ∈ A C) ↔ (y ∈ B ∨ y ∈ ∩x ∈ A C))
12	8, 10, 11	3bitr4 158	. 2 ⊢ (y ∈ ∩x ∈ A (B ∪ C) ↔ y ∈ (B ∪ ∩x ∈ A C))
13	12	cleqri 1101	1 ⊢ ∩x ∈ A (B ∪ C) = (B ∪ ∩x ∈ A C)

Syntax hints:   ↔ wb 127   ∨ wo 195   = wceq 1091   ∈ wcel 1092  ∀wral 1201  Vcvv 1348   ∪ cun 1485  ∩ciin 1995

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-ral 1205  df-v 1349  df-un 1490  df-iin 1997

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