Theorem unisngl 23470

Description: Taking the union of the set of singletons recovers the initial set. (Contributed by Thierry Arnoux, 9-Jan-2020.)

Hypothesis

Ref	Expression
dissnref.c	⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}

Assertion

Ref	Expression
unisngl	⊢ 𝑋 = ∪ 𝐶

Distinct variable groups:   𝑢,𝐶,𝑥   𝑢,𝑋,𝑥

Proof of Theorem unisngl

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dissnref.c	. . 3 ⊢ 𝐶 = {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
2	1	unieqi 4900	. 2 ⊢ ∪ 𝐶 = ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
3		simpl 482	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑦 ∈ 𝑢)
4		simpr 484	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
5	3, 4	eleqtrd 2837	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
6	5	exlimiv 1930	. . . . . . 7 ⊢ (∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
7		eqid 2736	. . . . . . . 8 ⊢ {𝑥} = {𝑥}
8		vsnex 5409	. . . . . . . . 9 ⊢ {𝑥} ∈ V
9		eleq2 2824	. . . . . . . . . 10 ⊢ (𝑢 = {𝑥} → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ {𝑥}))
10		eqeq1 2740	. . . . . . . . . 10 ⊢ (𝑢 = {𝑥} → (𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
11	9, 10	anbi12d 632	. . . . . . . . 9 ⊢ (𝑢 = {𝑥} → ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ (𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥})))
12	8, 11	spcev 3590	. . . . . . . 8 ⊢ ((𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥}) → ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
13	7, 12	mpan2 691	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} → ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
14	6, 13	impbii 209	. . . . . 6 ⊢ (∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ 𝑦 ∈ {𝑥})
15		velsn 4622	. . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
16		equcom 2018	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
17	14, 15, 16	3bitri 297	. . . . 5 ⊢ (∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ 𝑥 = 𝑦)
18	17	rexbii 3084	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ ∃𝑥 ∈ 𝑋 𝑥 = 𝑦)
19		r19.42v 3177	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ (𝑦 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}))
20	19	exbii 1848	. . . . 5 ⊢ (∃𝑢∃𝑥 ∈ 𝑋 (𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}))
21		rexcom4 3273	. . . . 5 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ ∃𝑢∃𝑥 ∈ 𝑋 (𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
22		eluniab 4902	. . . . 5 ⊢ (𝑦 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ↔ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}))
23	20, 21, 22	3bitr4ri 304	. . . 4 ⊢ (𝑦 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ↔ ∃𝑥 ∈ 𝑋 ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
24		risset 3221	. . . 4 ⊢ (𝑦 ∈ 𝑋 ↔ ∃𝑥 ∈ 𝑋 𝑥 = 𝑦)
25	18, 23, 24	3bitr4i 303	. . 3 ⊢ (𝑦 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ↔ 𝑦 ∈ 𝑋)
26	25	eqriv 2733	. 2 ⊢ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} = 𝑋
27	2, 26	eqtr2i 2760	1 ⊢ 𝑋 = ∪ 𝐶

Syntax hints:   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {cab 2714  ∃wrex 3061  {csn 4606  ∪ cuni 4888

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2715  df-cleq 2728  df-clel 2810  df-rex 3062  df-v 3466  df-un 3936  df-ss 3948  df-sn 4607  df-pr 4609  df-uni 4889

This theorem is referenced by:  dissnref  23471  dissnlocfin  23472