Theorem unisngl 23754

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 4879	. 2 ⊢ ∪ 𝐶 = ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}}
3		simpl 488	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑦 ∈ 𝑢)
4		simpr 490	. . . . . . . . 9 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑢 = {𝑥})
5	3, 4	eleqtrd 2862	. . . . . . . 8 ⊢ ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
6	5	exlimiv 1963	. . . . . . 7 ⊢ (∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
7		eqid 2760	. . . . . . . 8 ⊢ {𝑥} = {𝑥}
8		vsnex 5400	. . . . . . . . 9 ⊢ {𝑥} ∈ V
9		eleq2 2849	. . . . . . . . . 10 ⊢ (𝑢 = {𝑥} → (𝑦 ∈ 𝑢 ↔ 𝑦 ∈ {𝑥}))
10		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑢 = {𝑥} → (𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
11	9, 10	anbi12d 644	. . . . . . . . 9 ⊢ (𝑢 = {𝑥} → ((𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ (𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥})))
12	8, 11	spcev 3560	. . . . . . . 8 ⊢ ((𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥}) → ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
13	7, 12	mpan2 704	. . . . . . 7 ⊢ (𝑦 ∈ {𝑥} → ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
14	6, 13	impbii 212	. . . . . 6 ⊢ (∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ 𝑦 ∈ {𝑥})
15		velsn 4600	. . . . . 6 ⊢ (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
16		equcom 2051	. . . . . 6 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
17	14, 15, 16	3bitri 300	. . . . 5 ⊢ (∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ 𝑥 = 𝑦)
18	17	rexbii 3109	. . . 4 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ ∃𝑥 ∈ 𝑋 𝑥 = 𝑦)
19		r19.42v 3194	. . . . . 6 ⊢ (∃𝑥 ∈ 𝑋 (𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ (𝑦 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}))
20	19	exbii 1881	. . . . 5 ⊢ (∃𝑢∃𝑥 ∈ 𝑋 (𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}))
21		rexcom4 3289	. . . . 5 ⊢ (∃𝑥 ∈ 𝑋 ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}) ↔ ∃𝑢∃𝑥 ∈ 𝑋 (𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
22		eluniab 4881	. . . . 5 ⊢ (𝑦 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ↔ ∃𝑢(𝑦 ∈ 𝑢 ∧ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}))
23	20, 21, 22	3bitr4ri 307	. . . 4 ⊢ (𝑦 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ↔ ∃𝑥 ∈ 𝑋 ∃𝑢(𝑦 ∈ 𝑢 ∧ 𝑢 = {𝑥}))
24		risset 3237	. . . 4 ⊢ (𝑦 ∈ 𝑋 ↔ ∃𝑥 ∈ 𝑋 𝑥 = 𝑦)
25	18, 23, 24	3bitr4i 306	. . 3 ⊢ (𝑦 ∈ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} ↔ 𝑦 ∈ 𝑋)
26	25	eqriv 2757	. 2 ⊢ ∪ {𝑢 ∣ ∃𝑥 ∈ 𝑋 𝑢 = {𝑥}} = 𝑋
27	2, 26	eqtr2i 2784	1 ⊢ 𝑋 = ∪ 𝐶

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2145  {cab 2738  ∃wrex 3086  {csn 4584  ∪ cuni 4867

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-rex 3087  df-v 3452  df-un 3904  df-ss 3916  df-sn 4585  df-pr 4587  df-uni 4868

This theorem is used by:  dissnref  23755  dissnlocfin  23756