Theorem uniin 4897

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8796 for a condition where equality holds. (Contributed by NM, 4-Dec-2003.) (Proof shortened by Andrew Salmon, 29-Jun-2011.) (Proof shortened by Umit Teoman Dogan, 10-Jun-2026.)

Assertion

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

Proof of Theorem uniin

Step	Hyp	Ref	Expression
1		inss1 4190	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	unissi 4882	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐴
3		inss2 4191	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	unissi 4882	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐵
5	2, 4	ssini 4193	1 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)

Syntax hints:   ∩ cin 3905   ⊆ wss 3906  ∪ cuni 4873

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1573  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3417  df-v 3457  df-in 3913  df-ss 3923  df-uni 4874

This theorem is referenced by:  uniinqs  8796  psss  18637  tgval  23093  mapdunirnN  42405