Theorem undi 3503

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.)

Assertion

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

Proof of Theorem undi

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elin 3220	. . . 4 ⊢ (x ∈ (B ∩ C) ↔ (x ∈ B ∧ x ∈ C))
2	1	orbi2i 505	. . 3 ⊢ ((x ∈ A ∨ x ∈ (B ∩ C)) ↔ (x ∈ A ∨ (x ∈ B ∧ x ∈ C)))
3		ordi 834	. . 3 ⊢ ((x ∈ A ∨ (x ∈ B ∧ x ∈ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)))
4		elin 3220	. . . 4 ⊢ (x ∈ ((A ∪ B) ∩ (A ∪ C)) ↔ (x ∈ (A ∪ B) ∧ x ∈ (A ∪ C)))
5		elun 3221	. . . . 5 ⊢ (x ∈ (A ∪ B) ↔ (x ∈ A ∨ x ∈ B))
6		elun 3221	. . . . 5 ⊢ (x ∈ (A ∪ C) ↔ (x ∈ A ∨ x ∈ C))
7	5, 6	anbi12i 678	. . . 4 ⊢ ((x ∈ (A ∪ B) ∧ x ∈ (A ∪ C)) ↔ ((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)))
8	4, 7	bitr2i 241	. . 3 ⊢ (((x ∈ A ∨ x ∈ B) ∧ (x ∈ A ∨ x ∈ C)) ↔ x ∈ ((A ∪ B) ∩ (A ∪ C)))
9	2, 3, 8	3bitri 262	. 2 ⊢ ((x ∈ A ∨ x ∈ (B ∩ C)) ↔ x ∈ ((A ∪ B) ∩ (A ∪ C)))
10	9	uneqri 3407	1 ⊢ (A ∪ (B ∩ C)) = ((A ∪ B) ∩ (A ∪ C))

This proof depends on syntax axioms:   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710   ∪ cun 3208   ∩ cin 3209

This proof depends on 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 proof depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  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-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215

This theorem is used by:  undir  3505  dfif4  3674  dfif5  3675