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Theorem unisngl 23030
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 ∣ ∃𝑥𝑋 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.
StepHypRef Expression
1 dissnref.c . . 3 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
21unieqi 4921 . 2 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
3 simpl 483 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦𝑢)
4 simpr 485 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑢 = {𝑥})
53, 4eleqtrd 2835 . . . . . . . 8 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
65exlimiv 1933 . . . . . . 7 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
7 eqid 2732 . . . . . . . 8 {𝑥} = {𝑥}
8 vsnex 5429 . . . . . . . . 9 {𝑥} ∈ V
9 eleq2 2822 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑦𝑢𝑦 ∈ {𝑥}))
10 eqeq1 2736 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
119, 10anbi12d 631 . . . . . . . . 9 (𝑢 = {𝑥} → ((𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥})))
128, 11spcev 3596 . . . . . . . 8 ((𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥}) → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
137, 12mpan2 689 . . . . . . 7 (𝑦 ∈ {𝑥} → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
146, 13impbii 208 . . . . . 6 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑦 ∈ {𝑥})
15 velsn 4644 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
16 equcom 2021 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
1714, 15, 163bitri 296 . . . . 5 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑥 = 𝑦)
1817rexbii 3094 . . . 4 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑥𝑋 𝑥 = 𝑦)
19 r19.42v 3190 . . . . . 6 (∃𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2019exbii 1850 . . . . 5 (∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
21 rexcom4 3285 . . . . 5 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}))
22 eluniab 4923 . . . . 5 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2320, 21, 223bitr4ri 303 . . . 4 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}))
24 risset 3230 . . . 4 (𝑦𝑋 ↔ ∃𝑥𝑋 𝑥 = 𝑦)
2518, 23, 243bitr4i 302 . . 3 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ 𝑦𝑋)
2625eqriv 2729 . 2 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = 𝑋
272, 26eqtr2i 2761 1 𝑋 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 396   = wceq 1541  wex 1781  wcel 2106  {cab 2709  wrex 3070  {csn 4628   cuni 4908
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-pr 5427
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-rex 3071  df-v 3476  df-un 3953  df-in 3955  df-ss 3965  df-sn 4629  df-pr 4631  df-uni 4909
This theorem is referenced by:  dissnref  23031  dissnlocfin  23032
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