Theorem undi

Description: Distributive law for union over intersection. Exercise 11 of TakeutiZaring p. 17. (Contributed by NM, 30-Sep-2002) (Proof shortened by Andrew Salmon, 26-Jun-2011)

	Ref	Expression
Assertion	undi	⊢ ( 𝐴 ∪ ( 𝐵 ∩ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∩ ( 𝐴 ∪ 𝐶 ) )

Proof

Step	Hyp	Ref	Expression
1		elin	⊢ ( 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ↔ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) )
2	1	orbi2i	⊢ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) ↔ ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) )
3		ordi	⊢ ( ( 𝑥 ∈ 𝐴 ∨ ( 𝑥 ∈ 𝐵 ∧ 𝑥 ∈ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶 ) ) )
4		elin	⊢ ( 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∩ ( 𝐴 ∪ 𝐶 ) ) ↔ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ 𝑥 ∈ ( 𝐴 ∪ 𝐶 ) ) )
5		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) )
6		elun	⊢ ( 𝑥 ∈ ( 𝐴 ∪ 𝐶 ) ↔ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶 ) )
7	5 6	anbi12i	⊢ ( ( 𝑥 ∈ ( 𝐴 ∪ 𝐵 ) ∧ 𝑥 ∈ ( 𝐴 ∪ 𝐶 ) ) ↔ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶 ) ) )
8	4 7	bitr2i	⊢ ( ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵 ) ∧ ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐶 ) ) ↔ 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∩ ( 𝐴 ∪ 𝐶 ) ) )
9	2 3 8	3bitri	⊢ ( ( 𝑥 ∈ 𝐴 ∨ 𝑥 ∈ ( 𝐵 ∩ 𝐶 ) ) ↔ 𝑥 ∈ ( ( 𝐴 ∪ 𝐵 ) ∩ ( 𝐴 ∪ 𝐶 ) ) )
10	9	uneqri	⊢ ( 𝐴 ∪ ( 𝐵 ∩ 𝐶 ) ) = ( ( 𝐴 ∪ 𝐵 ) ∩ ( 𝐴 ∪ 𝐶 ) )