Theorem uniin 4879

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8763 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 4179	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	unissi 4864	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐴
3		inss2 4180	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	unissi 4864	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐵
5	2, 4	ssini 4182	1 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)

Syntax hints:   ∩ cin 3894   ⊆ wss 3895  ∪ cuni 4855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1553  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-rab 3405  df-v 3446  df-in 3902  df-ss 3912  df-uni 4856

This theorem is referenced by:  uniinqs  8763  psss  18584  tgval  22984  mapdunirnN  42212