Theorem iunxun 4048

Description: Separate a union in the index of an indexed union. (Contributed by NM, 26-Mar-2004.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

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

Proof of Theorem iunxun

Dummy variable y is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexun 3444	. . . 4 ⊢ (∃x ∈ (A ∪ B)y ∈ C ↔ (∃x ∈ A y ∈ C ∨ ∃x ∈ B y ∈ C))
2		eliun 3974	. . . . 5 ⊢ (y ∈ ∪x ∈ A C ↔ ∃x ∈ A y ∈ C)
3		eliun 3974	. . . . 5 ⊢ (y ∈ ∪x ∈ B C ↔ ∃x ∈ B y ∈ C)
4	2, 3	orbi12i 507	. . . 4 ⊢ ((y ∈ ∪x ∈ A C ∨ y ∈ ∪x ∈ B C) ↔ (∃x ∈ A y ∈ C ∨ ∃x ∈ B y ∈ C))
5	1, 4	bitr4i 243	. . 3 ⊢ (∃x ∈ (A ∪ B)y ∈ C ↔ (y ∈ ∪x ∈ A C ∨ y ∈ ∪x ∈ B C))
6		eliun 3974	. . 3 ⊢ (y ∈ ∪x ∈ (A ∪ B)C ↔ ∃x ∈ (A ∪ B)y ∈ C)
7		elun 3221	. . 3 ⊢ (y ∈ (∪x ∈ A C ∪ ∪x ∈ B C) ↔ (y ∈ ∪x ∈ A C ∨ y ∈ ∪x ∈ B C))
8	5, 6, 7	3bitr4i 268	. 2 ⊢ (y ∈ ∪x ∈ (A ∪ B)C ↔ y ∈ (∪x ∈ A C ∪ ∪x ∈ B C))
9	8	eqriv 2350	1 ⊢ ∪x ∈ (A ∪ B)C = (∪x ∈ A C ∪ ∪x ∈ B 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: (None)