Theorem unundi 4151

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

Assertion

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

Proof of Theorem unundi

Step	Hyp	Ref	Expression
1		unidm 4132	. . 3 ⊢ (𝐴 ∪ 𝐴) = 𝐴
2	1	uneq1i 4139	. 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = (𝐴 ∪ (𝐵 ∪ 𝐶))
3		un4 4150	. 2 ⊢ ((𝐴 ∪ 𝐴) ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))
4	2, 3	eqtr3i 2760	1 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = ((𝐴 ∪ 𝐵) ∪ (𝐴 ∪ 𝐶))

Syntax hints:   = wceq 1540   ∪ cun 3924

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 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-un 3931

This theorem is referenced by:  dfif5  4517