Theorem iunun 4047

Description: Separate a union in an indexed union. (Contributed by NM, 27-Dec-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

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

Proof of Theorem iunun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.43 2767	. . . 4 ⊢ (∃x ∈ A (y ∈ B ∨ y ∈ C) ↔ (∃x ∈ A y ∈ B ∨ ∃x ∈ A y ∈ C))
2		elun 3221	. . . . 5 ⊢ (y ∈ (B ∪ C) ↔ (y ∈ B ∨ y ∈ C))
3	2	rexbii 2640	. . . 4 ⊢ (∃x ∈ A y ∈ (B ∪ C) ↔ ∃x ∈ A (y ∈ B ∨ y ∈ C))
4		eliun 3974	. . . . 5 ⊢ (y ∈ ∪x ∈ A B ↔ ∃x ∈ A y ∈ B)
5		eliun 3974	. . . . 5 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
6	4, 5	orbi12i 507	. . . 4 ⊢ ((y ∈ ∪x ∈ A B ∨ y ∈ ∪x ∈ A C) ↔ (∃x ∈ A y ∈ B ∨ ∃x ∈ A y ∈ C))
7	1, 3, 6	3bitr4i 268	. . 3 ⊢ (∃x ∈ A y ∈ (B ∪ C) ↔ (y ∈ ∪x ∈ A B ∨ y ∈ ∪x ∈ A C))
8		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ A (B ∪ C) ↔ ∃x ∈ A y ∈ (B ∪ C))
9		elun 3221	. . 3 ⊢ (y ∈ (∪x ∈ A B ∪ ∪x ∈ A C) ↔ (y ∈ ∪x ∈ A B ∨ y ∈ ∪x ∈ A C))
10	7, 8, 9	3bitr4i 268	. 2 ⊢ (y ∈ ∪x ∈ A (B ∪ C) ↔ y ∈ (∪x ∈ A B ∪ ∪x ∈ A C))
11	10	eqriv 2350	1 ⊢ ∪x ∈ A (B ∪ C) = (∪x ∈ A B ∪ ∪x ∈ A C)

Syntax hints:   ∨ wo 357   = wceq 1642   ∈ wcel 1710  ∃wrex 2616   ∪ cun 3208  ∪ciun 3970

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-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-iun 3972

This theorem is referenced by:  iununi  4051