Theorem iinuni 4050

Description: A relationship involving union and indexed intersection. Exercise 23 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

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

Distinct variable groups:   x,A   x,B

Proof of Theorem iinuni

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 2758	. . . 4 ⊢ (∀x ∈ B (y ∈ A ∨ y ∈ x) ↔ (y ∈ A ∨ ∀x ∈ B y ∈ x))
2		elun 3221	. . . . 5 ⊢ (y ∈ (A ∪ x) ↔ (y ∈ A ∨ y ∈ x))
3	2	ralbii 2639	. . . 4 ⊢ (∀x ∈ B y ∈ (A ∪ x) ↔ ∀x ∈ B (y ∈ A ∨ y ∈ x))
4		vex 2863	. . . . . 6 ⊢ y ∈ V
5	4	elint2 3934	. . . . 5 ⊢ (y ∈ ∩B ↔ ∀x ∈ B y ∈ x)
6	5	orbi2i 505	. . . 4 ⊢ ((y ∈ A ∨ y ∈ ∩B) ↔ (y ∈ A ∨ ∀x ∈ B y ∈ x))
7	1, 3, 6	3bitr4ri 269	. . 3 ⊢ ((y ∈ A ∨ y ∈ ∩B) ↔ ∀x ∈ B y ∈ (A ∪ x))
8		elun 3221	. . 3 ⊢ (y ∈ (A ∪ ∩B) ↔ (y ∈ A ∨ y ∈ ∩B))
9		eliin 3975	. . . 4 ⊢ (y ∈ V → (y ∈ ∩x ∈ B (A ∪ x) ↔ ∀x ∈ B y ∈ (A ∪ x)))
10	4, 9	ax-mp 5	. . 3 ⊢ (y ∈ ∩x ∈ B (A ∪ x) ↔ ∀x ∈ B y ∈ (A ∪ x))
11	7, 8, 10	3bitr4i 268	. 2 ⊢ (y ∈ (A ∪ ∩B) ↔ y ∈ ∩x ∈ B (A ∪ x))
12	11	eqriv 2350	1 ⊢ (A ∪ ∩B) = ∩x ∈ B (A ∪ x)

Syntax hints:   ↔ wb 176   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∀wral 2615  Vcvv 2860   ∪ cun 3208  ∩cint 3927  ∩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-int 3928  df-iin 3973

This theorem is referenced by: (None)