Theorem iununi 5059

Description: A relationship involving union and indexed union. Exercise 25 of [Enderton] p. 33. (Contributed by NM, 25-Nov-2003.) (Proof shortened by Mario Carneiro, 17-Nov-2016.)

Assertion

Ref	Expression
iununi	⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem iununi

Step	Hyp	Ref	Expression
1		df-ne 2957	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2		iunconst 4961	. . . . . . 7 ⊢ (𝐵 ≠ ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
3	1, 2	sylbir 238	. . . . . 6 ⊢ (¬ 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
4		iun0 5020	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 ∅ = ∅
5		id 23	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	5	iuneq2d 4981	. . . . . . 7 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = ∪ 𝑥 ∈ 𝐵 ∅)
7	4, 6, 5	3eqtr4a 2822	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
8	3, 7	ja 188	. . . . 5 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
9	8	eqcomd 2767	. . . 4 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → 𝐴 = ∪ 𝑥 ∈ 𝐵 𝐴)
10	9	uneq1d 4114	. . 3 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥))
11		uniiun 5017	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
12	11	uneq2i 4112	. . 3 ⊢ (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
13		iunun 5053	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
14	10, 12, 13	3eqtr4g 2821	. 2 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))
15		unieq 4878	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
16		uni0 4896	. . . . . . 7 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2812	. . . . . 6 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∅)
18	17	uneq2d 4115	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∅))
19		un0 4344	. . . . 5 ⊢ (𝐴 ∪ ∅) = 𝐴
20	18, 19	eqtrdi 2812	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = 𝐴)
21		iuneq1 4968	. . . . 5 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥))
22		0iun 5021	. . . . 5 ⊢ ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥) = ∅
23	21, 22	eqtrdi 2812	. . . 4 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∅)
24	20, 23	eqeq12d 2777	. . 3 ⊢ (𝐵 = ∅ → ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) ↔ 𝐴 = ∅))
25	24	biimpcd 252	. 2 ⊢ ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
26	14, 25	impbii 212	1 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   = wceq 1570   ≠ wne 2956   ∪ cun 3897  ∅c0 4279  ∪ cuni 4867  ∪ ciun 4951

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-uni 4868  df-iun 4953

This theorem is used by: (None)