Theorem iununi 4051

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 2519	. . . . . . 7 ⊢ (B ≠ ∅ ↔ ¬ B = ∅)
2		iunconst 3978	. . . . . . 7 ⊢ (B ≠ ∅ → ∪x ∈ B A = A)
3	1, 2	sylbir 204	. . . . . 6 ⊢ (¬ B = ∅ → ∪x ∈ B A = A)
4		iun0 4023	. . . . . . 7 ⊢ ∪x ∈ B ∅ = ∅
5		id 19	. . . . . . . 8 ⊢ (A = ∅ → A = ∅)
6	5	iuneq2d 3995	. . . . . . 7 ⊢ (A = ∅ → ∪x ∈ B A = ∪x ∈ B ∅)
7	4, 6, 5	3eqtr4a 2411	. . . . . 6 ⊢ (A = ∅ → ∪x ∈ B A = A)
8	3, 7	ja 153	. . . . 5 ⊢ ((B = ∅ → A = ∅) → ∪x ∈ B A = A)
9	8	eqcomd 2358	. . . 4 ⊢ ((B = ∅ → A = ∅) → A = ∪x ∈ B A)
10	9	uneq1d 3418	. . 3 ⊢ ((B = ∅ → A = ∅) → (A ∪ ∪x ∈ B x) = (∪x ∈ B A ∪ ∪x ∈ B x))
11		uniiun 4020	. . . 4 ⊢ ∪B = ∪x ∈ B x
12	11	uneq2i 3416	. . 3 ⊢ (A ∪ ∪B) = (A ∪ ∪x ∈ B x)
13		iunun 4047	. . 3 ⊢ ∪x ∈ B (A ∪ x) = (∪x ∈ B A ∪ ∪x ∈ B x)
14	10, 12, 13	3eqtr4g 2410	. 2 ⊢ ((B = ∅ → A = ∅) → (A ∪ ∪B) = ∪x ∈ B (A ∪ x))
15		unieq 3901	. . . . . . 7 ⊢ (B = ∅ → ∪B = ∪∅)
16		uni0 3919	. . . . . . 7 ⊢ ∪∅ = ∅
17	15, 16	syl6eq 2401	. . . . . 6 ⊢ (B = ∅ → ∪B = ∅)
18	17	uneq2d 3419	. . . . 5 ⊢ (B = ∅ → (A ∪ ∪B) = (A ∪ ∅))
19		un0 3576	. . . . 5 ⊢ (A ∪ ∅) = A
20	18, 19	syl6eq 2401	. . . 4 ⊢ (B = ∅ → (A ∪ ∪B) = A)
21		iuneq1 3983	. . . . 5 ⊢ (B = ∅ → ∪x ∈ B (A ∪ x) = ∪x ∈ ∅ (A ∪ x))
22		0iun 4024	. . . . 5 ⊢ ∪x ∈ ∅ (A ∪ x) = ∅
23	21, 22	syl6eq 2401	. . . 4 ⊢ (B = ∅ → ∪x ∈ B (A ∪ x) = ∅)
24	20, 23	eqeq12d 2367	. . 3 ⊢ (B = ∅ → ((A ∪ ∪B) = ∪x ∈ B (A ∪ x) ↔ A = ∅))
25	24	biimpcd 215	. 2 ⊢ ((A ∪ ∪B) = ∪x ∈ B (A ∪ x) → (B = ∅ → A = ∅))
26	14, 25	impbii 180	1 ⊢ ((B = ∅ → A = ∅) ↔ (A ∪ ∪B) = ∪x ∈ B (A ∪ x))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 176   = wceq 1642   ≠ wne 2517   ∪ cun 3208  ∅c0 3551  ∪cuni 3892  ∪ciun 3970

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-uni 3893  df-iun 3972

This theorem is referenced by: (None)