Theorem uniin 3912

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs in set.mm for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.)

Assertion

Ref	Expression
uniin	⊢ ∪(A ∩ B) ⊆ (∪A ∩ ∪B)

Proof of Theorem uniin

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		19.40 1609	. . . 4 ⊢ (∃y((x ∈ y ∧ y ∈ A) ∧ (x ∈ y ∧ y ∈ B)) → (∃y(x ∈ y ∧ y ∈ A) ∧ ∃y(x ∈ y ∧ y ∈ B)))
2		elin 3220	. . . . . . 7 ⊢ (y ∈ (A ∩ B) ↔ (y ∈ A ∧ y ∈ B))
3	2	anbi2i 675	. . . . . 6 ⊢ ((x ∈ y ∧ y ∈ (A ∩ B)) ↔ (x ∈ y ∧ (y ∈ A ∧ y ∈ B)))
4		anandi 801	. . . . . 6 ⊢ ((x ∈ y ∧ (y ∈ A ∧ y ∈ B)) ↔ ((x ∈ y ∧ y ∈ A) ∧ (x ∈ y ∧ y ∈ B)))
5	3, 4	bitri 240	. . . . 5 ⊢ ((x ∈ y ∧ y ∈ (A ∩ B)) ↔ ((x ∈ y ∧ y ∈ A) ∧ (x ∈ y ∧ y ∈ B)))
6	5	exbii 1582	. . . 4 ⊢ (∃y(x ∈ y ∧ y ∈ (A ∩ B)) ↔ ∃y((x ∈ y ∧ y ∈ A) ∧ (x ∈ y ∧ y ∈ B)))
7		eluni 3895	. . . . 5 ⊢ (x ∈ ∪A ↔ ∃y(x ∈ y ∧ y ∈ A))
8		eluni 3895	. . . . 5 ⊢ (x ∈ ∪B ↔ ∃y(x ∈ y ∧ y ∈ B))
9	7, 8	anbi12i 678	. . . 4 ⊢ ((x ∈ ∪A ∧ x ∈ ∪B) ↔ (∃y(x ∈ y ∧ y ∈ A) ∧ ∃y(x ∈ y ∧ y ∈ B)))
10	1, 6, 9	3imtr4i 257	. . 3 ⊢ (∃y(x ∈ y ∧ y ∈ (A ∩ B)) → (x ∈ ∪A ∧ x ∈ ∪B))
11		eluni 3895	. . 3 ⊢ (x ∈ ∪(A ∩ B) ↔ ∃y(x ∈ y ∧ y ∈ (A ∩ B)))
12		elin 3220	. . 3 ⊢ (x ∈ (∪A ∩ ∪B) ↔ (x ∈ ∪A ∧ x ∈ ∪B))
13	10, 11, 12	3imtr4i 257	. 2 ⊢ (x ∈ ∪(A ∩ B) → x ∈ (∪A ∩ ∪B))
14	13	ssriv 3278	1 ⊢ ∪(A ∩ B) ⊆ (∪A ∩ ∪B)

Syntax hints:   ∧ wa 358  ∃wex 1541   ∈ wcel 1710   ∩ cin 3209   ⊆ wss 3258  ∪cuni 3892

This theorem was proved from 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 theorem 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-ss 3260  df-uni 3893

This theorem is referenced by: (None)