Theorem iinun2 4033

Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 4021 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Assertion

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

Distinct variable group:   x,B

Allowed substitution hints:   A(x)   C(x)

Proof of Theorem iinun2

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 2758	. . . 4 ⊢ (∀x ∈ A (y ∈ B ∨ y ∈ C) ↔ (y ∈ B ∨ ∀x ∈ A y ∈ C))
2		elun 3221	. . . . 5 ⊢ (y ∈ (B ∪ C) ↔ (y ∈ B ∨ y ∈ C))
3	2	ralbii 2639	. . . 4 ⊢ (∀x ∈ A y ∈ (B ∪ C) ↔ ∀x ∈ A (y ∈ B ∨ y ∈ C))
4		vex 2863	. . . . . 6 ⊢ y ∈ V
5		eliin 3975	. . . . . 6 ⊢ (y ∈ V → (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C))
6	4, 5	ax-mp 5	. . . . 5 ⊢ (y ∈ ∩x ∈ A C ↔ ∀x ∈ A y ∈ C)
7	6	orbi2i 505	. . . 4 ⊢ ((y ∈ B ∨ y ∈ ∩x ∈ A C) ↔ (y ∈ B ∨ ∀x ∈ A y ∈ C))
8	1, 3, 7	3bitr4i 268	. . 3 ⊢ (∀x ∈ A y ∈ (B ∪ C) ↔ (y ∈ B ∨ y ∈ ∩x ∈ A C))
9		eliin 3975	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ A (B ∪ C) ↔ ∀x ∈ A y ∈ (B ∪ C)))
10	4, 9	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ A (B ∪ C) ↔ ∀x ∈ A y ∈ (B ∪ C))
11		elun 3221	. . 3 ⊢ (y ∈ (B ∪ ∩x ∈ A C) ↔ (y ∈ B ∨ y ∈ ∩x ∈ A C))
12	8, 10, 11	3bitr4i 268	. 2 ⊢ (y ∈ ∩x ∈ A (B ∪ C) ↔ y ∈ (B ∪ ∩x ∈ A C))
13	12	eqriv 2350	1 ⊢ ∩x ∈ A (B ∪ C) = (B ∪ ∩x ∈ A C)

Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ∪ cun 3208  ∩ciin 3971

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-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-iin 3973

This theorem is referenced by: (None)