Theorem unundir 4157

Description: Union distributes over itself. (Contributed by NM, 17-Aug-2004.)

Assertion

Ref	Expression
unundir	⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))

Proof of Theorem unundir

Step	Hyp	Ref	Expression
1		unidm 4137	. . 3 ⊢ (𝐶 ∪ 𝐶) = 𝐶
2	1	uneq2i 4145	. 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)
3		un4 4155	. 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐶)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))
4	2, 3	eqtr3i 2761	1 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))

Syntax hints:   = wceq 1540   ∪ cun 3929

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-v 3466  df-un 3936

This theorem is referenced by:  iocunico  43210