Theorem unundir 4105

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 4086	. . 3 ⊢ (𝐶 ∪ 𝐶) = 𝐶
2	1	uneq2i 4094	. 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ 𝐶)
3		un4 4103	. 2 ⊢ ((𝐴 ∪ 𝐵) ∪ (𝐶 ∪ 𝐶)) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))
4	2, 3	eqtr3i 2768	1 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ (𝐵 ∪ 𝐶))

Syntax hints:   = wceq 1539   ∪ cun 3885

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-un 3892

This theorem is referenced by:  iocunico  41042