Theorem uniun

Description: The class union of the union of two classes. Theorem 8.3 of Quine p. 53. (Contributed by NM, 20-Aug-1993)

	Ref	Expression
Assertion	uniun	⊢ ∪ ( 𝐴 ∪ 𝐵 ) = ( ∪ 𝐴 ∪ ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		19.43	⊢ ( ∃ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∨ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) ↔ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∨ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
2		elun	⊢ ( 𝑦 ∈ ( 𝐴 ∪ 𝐵 ) ↔ ( 𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵 ) )
3	2	anbi2i	⊢ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∪ 𝐵 ) ) ↔ ( 𝑥 ∈ 𝑦 ∧ ( 𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵 ) ) )
4		andi	⊢ ( ( 𝑥 ∈ 𝑦 ∧ ( 𝑦 ∈ 𝐴 ∨ 𝑦 ∈ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∨ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
5	3 4	bitri	⊢ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∪ 𝐵 ) ) ↔ ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∨ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
6	5	exbii	⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∪ 𝐵 ) ) ↔ ∃ 𝑦 ( ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∨ ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
7		eluni	⊢ ( 𝑥 ∈ ∪ 𝐴 ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) )
8		eluni	⊢ ( 𝑥 ∈ ∪ 𝐵 ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) )
9	7 8	orbi12i	⊢ ( ( 𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵 ) ↔ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐴 ) ∨ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝐵 ) ) )
10	1 6 9	3bitr4i	⊢ ( ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∪ 𝐵 ) ) ↔ ( 𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵 ) )
11		eluni	⊢ ( 𝑥 ∈ ∪ ( 𝐴 ∪ 𝐵 ) ↔ ∃ 𝑦 ( 𝑥 ∈ 𝑦 ∧ 𝑦 ∈ ( 𝐴 ∪ 𝐵 ) ) )
12		elun	⊢ ( 𝑥 ∈ ( ∪ 𝐴 ∪ ∪ 𝐵 ) ↔ ( 𝑥 ∈ ∪ 𝐴 ∨ 𝑥 ∈ ∪ 𝐵 ) )
13	10 11 12	3bitr4i	⊢ ( 𝑥 ∈ ∪ ( 𝐴 ∪ 𝐵 ) ↔ 𝑥 ∈ ( ∪ 𝐴 ∪ ∪ 𝐵 ) )
14	13	eqriv	⊢ ∪ ( 𝐴 ∪ 𝐵 ) = ( ∪ 𝐴 ∪ ∪ 𝐵 )