Theorem uniin 4931

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8837 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 1886	. . . 4 ⊢ (∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)) → (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
2		elin 3967	. . . . . . 7 ⊢ (𝑦 ∈ (𝐴 ∩ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵))
3	2	anbi2i 623	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ (𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
4		anandi 676	. . . . . 6 ⊢ ((𝑥 ∈ 𝑦 ∧ (𝑦 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
5	3, 4	bitri 275	. . . . 5 ⊢ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
6	5	exbii 1848	. . . 4 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) ↔ ∃𝑦((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
7		eluni 4910	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐴 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴))
8		eluni 4910	. . . . 5 ⊢ (𝑥 ∈ ∪ 𝐵 ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵))
9	7, 8	anbi12i 628	. . . 4 ⊢ ((𝑥 ∈ ∪ 𝐴 ∧ 𝑥 ∈ ∪ 𝐵) ↔ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴) ∧ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵)))
10	1, 6, 9	3imtr4i 292	. . 3 ⊢ (∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)) → (𝑥 ∈ ∪ 𝐴 ∧ 𝑥 ∈ ∪ 𝐵))
11		eluni 4910	. . 3 ⊢ (𝑥 ∈ ∪ (𝐴 ∩ 𝐵) ↔ ∃𝑦(𝑥 ∈ 𝑦 ∧ 𝑦 ∈ (𝐴 ∩ 𝐵)))
12		elin 3967	. . 3 ⊢ (𝑥 ∈ (∪ 𝐴 ∩ ∪ 𝐵) ↔ (𝑥 ∈ ∪ 𝐴 ∧ 𝑥 ∈ ∪ 𝐵))
13	10, 11, 12	3imtr4i 292	. 2 ⊢ (𝑥 ∈ ∪ (𝐴 ∩ 𝐵) → 𝑥 ∈ (∪ 𝐴 ∩ ∪ 𝐵))
14	13	ssriv 3987	1 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)

Syntax hints:   ∧ wa 395  ∃wex 1779   ∈ wcel 2108   ∩ cin 3950   ⊆ wss 3951  ∪ cuni 4907

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 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-in 3958  df-ss 3968  df-uni 4908

This theorem is referenced by:  uniinqs  8837  psss  18625  tgval  22962  mapdunirnN  41652