Theorem uniin 4896

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8856 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 4136	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	unissi 4881	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐴
3		inss2 4137	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	unissi 4881	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐵
5	2, 4	ssini 4139	1 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)

This proof depends on syntax axioms:   ⊆ wss 3900   ∩ cin 4092  ∪ cuni 4872

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-in 4094  df-uni 4873

This theorem is used by:  uniinqs  8856  psss  18804  tgval  23328  mapdunirnN  42973