Theorem iinun2 5030

Description: Indexed intersection of union. Generalization of half of theorem "Distributive laws" in [Enderton] p. 30. Use intiin 5017 to recover Enderton's theorem. (Contributed by NM, 19-Aug-2004.)

Assertion

Ref	Expression
iinun2	⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶)

Distinct variable group:   𝑥,𝐵

Allowed substitution hints:   𝐴(𝑥)   𝐶(𝑥)

Proof of Theorem iinun2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.32v 3195	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
2		elun 4151	. . . . 5 ⊢ (𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
3	2	ralbii 3108	. . . 4 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ 𝐶))
4		eliin 4955	. . . . . 6 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
5	4	elv 3455	. . . . 5 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶)
6	5	orbi2i 926	. . . 4 ⊢ ((𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ ∀𝑥 ∈ 𝐴 𝑦 ∈ 𝐶))
7	1, 3, 6	3bitr4i 306	. . 3 ⊢ (∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
8		eliin 4955	. . . 4 ⊢ (𝑦 ∈ V → (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶)))
9	8	elv 3455	. . 3 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ ∀𝑥 ∈ 𝐴 𝑦 ∈ (𝐵 ∪ 𝐶))
10		elun 4151	. . 3 ⊢ (𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶) ↔ (𝑦 ∈ 𝐵 ∨ 𝑦 ∈ ∩ 𝑥 ∈ 𝐴 𝐶))
11	7, 9, 10	3bitr4i 306	. 2 ⊢ (𝑦 ∈ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) ↔ 𝑦 ∈ (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶))
12	11	eqriv 2757	1 ⊢ ∩ 𝑥 ∈ 𝐴 (𝐵 ∪ 𝐶) = (𝐵 ∪ ∩ 𝑥 ∈ 𝐴 𝐶)

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1569   ∈ wcel 2144  ∀wral 3076  Vcvv 3450   ∪ cun 4148  ∩ ciin 4951

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-or 862  df-tru 1572  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-v 3452  df-un 4150  df-iin 4953

This theorem is used by: (None)