Theorem iuncom 3976

Description: Commutation of indexed unions. (Contributed by NM, 18-Dec-2008.)

Assertion

Ref	Expression
iuncom	⊢ ∪x ∈ A ∪y ∈ B C = ∪y ∈ B ∪x ∈ A C

Distinct variable groups:   y,A   x,B   x,y

Allowed substitution hints:   A(x)   B(y)   C(x,y)

Proof of Theorem iuncom

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		rexcom 2773	. . . 4 ⊢ (∃x ∈ A ∃y ∈ B z ∈ C ↔ ∃y ∈ B ∃x ∈ A z ∈ C)
2		eliun 3974	. . . . 5 ⊢ (z ∈ ∪y ∈ B C ↔ ∃y ∈ B z ∈ C)
3	2	rexbii 2640	. . . 4 ⊢ (∃x ∈ A z ∈ ∪y ∈ B C ↔ ∃x ∈ A ∃y ∈ B z ∈ C)
4		eliun 3974	. . . . 5 ⊢ (z ∈ ∪x ∈ A C ↔ ∃x ∈ A z ∈ C)
5	4	rexbii 2640	. . . 4 ⊢ (∃y ∈ B z ∈ ∪x ∈ A C ↔ ∃y ∈ B ∃x ∈ A z ∈ C)
6	1, 3, 5	3bitr4i 268	. . 3 ⊢ (∃x ∈ A z ∈ ∪y ∈ B C ↔ ∃y ∈ B z ∈ ∪x ∈ A C)
7		eliun 3974	. . 3 ⊢ (z ∈ ∪x ∈ A ∪y ∈ B C ↔ ∃x ∈ A z ∈ ∪y ∈ B C)
8		eliun 3974	. . 3 ⊢ (z ∈ ∪y ∈ B ∪x ∈ A C ↔ ∃y ∈ B z ∈ ∪x ∈ A C)
9	6, 7, 8	3bitr4i 268	. 2 ⊢ (z ∈ ∪x ∈ A ∪y ∈ B C ↔ z ∈ ∪y ∈ B ∪x ∈ A C)
10	9	eqriv 2350	1 ⊢ ∪x ∈ A ∪y ∈ B C = ∪y ∈ B ∪x ∈ A C

Syntax hints:   = wceq 1642   ∈ wcel 1710  ∃wrex 2616  ∪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-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-iun 3972

This theorem is referenced by: (None)