Theorem uniin 4891

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8783 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 4191	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	unissi 4876	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐴
3		inss2 4192	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	unissi 4876	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐵
5	2, 4	ssini 4194	1 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)

Syntax hints:   ∩ cin 3906   ⊆ wss 3907  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1566  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-rab 3418  df-v 3459  df-in 3914  df-ss 3924  df-uni 4868

This theorem is referenced by:  uniinqs  8783  psss  18624  tgval  23069  mapdunirnN  42281