Theorem uniin 4890

Description: The class union of the intersection of two classes. Exercise 4.12(n) of [Mendelson] p. 235. See uniinqs 8823 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 4132	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
2	1	unissi 4875	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐴
3		inss2 4133	. . 3 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐵
4	3	unissi 4875	. 2 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ ∪ 𝐵
5	2, 4	ssini 4135	1 ⊢ ∪ (𝐴 ∩ 𝐵) ⊆ (∪ 𝐴 ∩ ∪ 𝐵)

This proof depends on syntax axioms:   ⊆ wss 3896   ∩ cin 4088  ∪ cuni 4866

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-rab 3413  df-v 3452  df-ss 3897  df-in 4090  df-uni 4867

This theorem is used by:  uniinqs  8823  psss  18761  tgval  23280  mapdunirnN  42814