Theorem iununi 5122

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 2947	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2		iunconst 5024	. . . . . . 7 ⊢ (𝐵 ≠ ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
3	1, 2	sylbir 235	. . . . . 6 ⊢ (¬ 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
4		iun0 5085	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 ∅ = ∅
5		id 22	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	5	iuneq2d 5045	. . . . . . 7 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = ∪ 𝑥 ∈ 𝐵 ∅)
7	4, 6, 5	3eqtr4a 2806	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
8	3, 7	ja 186	. . . . 5 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
9	8	eqcomd 2746	. . . 4 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → 𝐴 = ∪ 𝑥 ∈ 𝐵 𝐴)
10	9	uneq1d 4190	. . 3 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥))
11		uniiun 5081	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
12	11	uneq2i 4188	. . 3 ⊢ (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
13		iunun 5116	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
14	10, 12, 13	3eqtr4g 2805	. 2 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))
15		unieq 4942	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
16		uni0 4959	. . . . . . 7 ⊢ ∪ ∅ = ∅
17	15, 16	eqtrdi 2796	. . . . . 6 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∅)
18	17	uneq2d 4191	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∅))
19		un0 4417	. . . . 5 ⊢ (𝐴 ∪ ∅) = 𝐴
20	18, 19	eqtrdi 2796	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = 𝐴)
21		iuneq1 5031	. . . . 5 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥))
22		0iun 5086	. . . . 5 ⊢ ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥) = ∅
23	21, 22	eqtrdi 2796	. . . 4 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∅)
24	20, 23	eqeq12d 2756	. . 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 1537   ≠ wne 2946   ∪ cun 3974  ∅c0 4352  ∪ cuni 4931  ∪ ciun 5015

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ne 2947  df-ral 3068  df-rex 3077  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-sn 4649  df-uni 4932  df-iun 5017

This theorem is referenced by: (None)