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Theorem unisngl 22901
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 4882 . 2 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
3 simpl 484 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦𝑢)
4 simpr 486 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑢 = {𝑥})
53, 4eleqtrd 2836 . . . . . . . 8 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
65exlimiv 1934 . . . . . . 7 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
7 eqid 2733 . . . . . . . 8 {𝑥} = {𝑥}
8 vsnex 5390 . . . . . . . . 9 {𝑥} ∈ V
9 eleq2 2823 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑦𝑢𝑦 ∈ {𝑥}))
10 eqeq1 2737 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
119, 10anbi12d 632 . . . . . . . . 9 (𝑢 = {𝑥} → ((𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥})))
128, 11spcev 3567 . . . . . . . 8 ((𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥}) → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
137, 12mpan2 690 . . . . . . 7 (𝑦 ∈ {𝑥} → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
146, 13impbii 208 . . . . . 6 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑦 ∈ {𝑥})
15 velsn 4606 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
16 equcom 2022 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
1714, 15, 163bitri 297 . . . . 5 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑥 = 𝑦)
1817rexbii 3094 . . . 4 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑥𝑋 𝑥 = 𝑦)
19 r19.42v 3184 . . . . . 6 (∃𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2019exbii 1851 . . . . 5 (∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
21 rexcom4 3270 . . . . 5 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}))
22 eluniab 4884 . . . . 5 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2320, 21, 223bitr4ri 304 . . . 4 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}))
24 risset 3220 . . . 4 (𝑦𝑋 ↔ ∃𝑥𝑋 𝑥 = 𝑦)
2518, 23, 243bitr4i 303 . . 3 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ 𝑦𝑋)
2625eqriv 2730 . 2 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = 𝑋
272, 26eqtr2i 2762 1 𝑋 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 397   = wceq 1542  wex 1782  wcel 2107  {cab 2710  wrex 3070  {csn 4590   cuni 4869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5260  ax-pr 5388
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rex 3071  df-v 3449  df-un 3919  df-in 3921  df-ss 3931  df-sn 4591  df-pr 4593  df-uni 4870
This theorem is referenced by:  dissnref  22902  dissnlocfin  22903
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