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Theorem unisngl 21613
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 4605 . 2 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
3 simpl 474 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦𝑢)
4 simpr 477 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑢 = {𝑥})
53, 4eleqtrd 2846 . . . . . . . 8 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
65exlimiv 2025 . . . . . . 7 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
7 eqid 2765 . . . . . . . 8 {𝑥} = {𝑥}
8 snex 5066 . . . . . . . . 9 {𝑥} ∈ V
9 eleq2 2833 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑦𝑢𝑦 ∈ {𝑥}))
10 eqeq1 2769 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
119, 10anbi12d 624 . . . . . . . . 9 (𝑢 = {𝑥} → ((𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥})))
128, 11spcev 3453 . . . . . . . 8 ((𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥}) → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
137, 12mpan2 682 . . . . . . 7 (𝑦 ∈ {𝑥} → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
146, 13impbii 200 . . . . . 6 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑦 ∈ {𝑥})
15 velsn 4352 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
16 equcom 2115 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
1714, 15, 163bitri 288 . . . . 5 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑥 = 𝑦)
1817rexbii 3188 . . . 4 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑥𝑋 𝑥 = 𝑦)
19 r19.42v 3239 . . . . . 6 (∃𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2019exbii 1943 . . . . 5 (∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
21 rexcom4 3378 . . . . 5 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}))
22 eluniab 4607 . . . . 5 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2320, 21, 223bitr4ri 295 . . . 4 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}))
24 risset 3209 . . . 4 (𝑦𝑋 ↔ ∃𝑥𝑋 𝑥 = 𝑦)
2518, 23, 243bitr4i 294 . . 3 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ 𝑦𝑋)
2625eqriv 2762 . 2 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = 𝑋
272, 26eqtr2i 2788 1 𝑋 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 384   = wceq 1652  wex 1874  wcel 2155  {cab 2751  wrex 3056  {csn 4336   cuni 4596
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-v 3352  df-dif 3737  df-un 3739  df-nul 4082  df-sn 4337  df-pr 4339  df-uni 4597
This theorem is referenced by:  dissnref  21614  dissnlocfin  21615
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