Theorem iununi 5098

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 2940	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2		iunconst 5000	. . . . . . 7 ⊢ (𝐵 ≠ ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
3	1, 2	sylbir 235	. . . . . 6 ⊢ (¬ 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
4		iun0 5061	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 ∅ = ∅
5		id 22	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	5	iuneq2d 5021	. . . . . . 7 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = ∪ 𝑥 ∈ 𝐵 ∅)
7	4, 6, 5	3eqtr4a 2802	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
8	3, 7	ja 186	. . . . 5 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
9	8	eqcomd 2742	. . . 4 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → 𝐴 = ∪ 𝑥 ∈ 𝐵 𝐴)
10	9	uneq1d 4166	. . 3 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥))
11		uniiun 5057	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
12	11	uneq2i 4164	. . 3 ⊢ (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
13		iunun 5092	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
14	10, 12, 13	3eqtr4g 2801	. 2 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))
15		unieq 4917	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
16		uni0 4934	. . . . . . 7 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2792	. . . . . 6 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∅)
18	17	uneq2d 4167	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∅))
19		un0 4393	. . . . 5 ⊢ (𝐴 ∪ ∅) = 𝐴
20	18, 19	eqtrdi 2792	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = 𝐴)
21		iuneq1 5007	. . . . 5 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥))
22		0iun 5062	. . . . 5 ⊢ ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥) = ∅
23	21, 22	eqtrdi 2792	. . . 4 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∅)
24	20, 23	eqeq12d 2752	. . 3 ⊢ (𝐵 = ∅ → ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) ↔ 𝐴 = ∅))
25	24	biimpcd 249	. 2 ⊢ ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
26	14, 25	impbii 209	1 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   = wceq 1539   ≠ wne 2939   ∪ cun 3948  ∅c0 4332  ∪ cuni 4906  ∪ ciun 4990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-sn 4626  df-uni 4907  df-iun 4992

This theorem is referenced by: (None)