Theorem iununi 4610

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 2795	. . . . . . 7 ⊢ (𝐵 ≠ ∅ ↔ ¬ 𝐵 = ∅)
2		iunconst 4529	. . . . . . 7 ⊢ (𝐵 ≠ ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
3	1, 2	sylbir 225	. . . . . 6 ⊢ (¬ 𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
4		iun0 4576	. . . . . . 7 ⊢ ∪ 𝑥 ∈ 𝐵 ∅ = ∅
5		id 22	. . . . . . . 8 ⊢ (𝐴 = ∅ → 𝐴 = ∅)
6	5	iuneq2d 4547	. . . . . . 7 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = ∪ 𝑥 ∈ 𝐵 ∅)
7	4, 6, 5	3eqtr4a 2682	. . . . . 6 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
8	3, 7	ja 173	. . . . 5 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → ∪ 𝑥 ∈ 𝐵 𝐴 = 𝐴)
9	8	eqcomd 2628	. . . 4 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → 𝐴 = ∪ 𝑥 ∈ 𝐵 𝐴)
10	9	uneq1d 3766	. . 3 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥))
11		uniiun 4573	. . . 4 ⊢ ∪ 𝐵 = ∪ 𝑥 ∈ 𝐵 𝑥
12	11	uneq2i 3764	. . 3 ⊢ (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
13		iunun 4604	. . 3 ⊢ ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = (∪ 𝑥 ∈ 𝐵 𝐴 ∪ ∪ 𝑥 ∈ 𝐵 𝑥)
14	10, 12, 13	3eqtr4g 2681	. 2 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) → (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))
15		unieq 4444	. . . . . . 7 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∪ ∅)
16		uni0 4465	. . . . . . 7 ⊢ ∪ ∅ = ∅
17	15, 16	syl6eq 2672	. . . . . 6 ⊢ (𝐵 = ∅ → ∪ 𝐵 = ∅)
18	17	uneq2d 3767	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = (𝐴 ∪ ∅))
19		un0 3967	. . . . 5 ⊢ (𝐴 ∪ ∅) = 𝐴
20	18, 19	syl6eq 2672	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 ∪ ∪ 𝐵) = 𝐴)
21		iuneq1 4534	. . . . 5 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥))
22		0iun 4577	. . . . 5 ⊢ ∪ 𝑥 ∈ ∅ (𝐴 ∪ 𝑥) = ∅
23	21, 22	syl6eq 2672	. . . 4 ⊢ (𝐵 = ∅ → ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) = ∅)
24	20, 23	eqeq12d 2637	. . 3 ⊢ (𝐵 = ∅ → ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) ↔ 𝐴 = ∅))
25	24	biimpcd 239	. 2 ⊢ ((𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥) → (𝐵 = ∅ → 𝐴 = ∅))
26	14, 25	impbii 199	1 ⊢ ((𝐵 = ∅ → 𝐴 = ∅) ↔ (𝐴 ∪ ∪ 𝐵) = ∪ 𝑥 ∈ 𝐵 (𝐴 ∪ 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   = wceq 1483   ≠ wne 2794   ∪ cun 3572  ∅c0 3915  ∪ cuni 4436  ∪ ciun 4520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-sn 4178  df-uni 4437  df-iun 4522

This theorem is referenced by: (None)