MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  unisngl Structured version   Visualization version   GIF version

Theorem unisngl 22424
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 4832 . 2 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
3 simpl 486 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦𝑢)
4 simpr 488 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑢 = {𝑥})
53, 4eleqtrd 2840 . . . . . . . 8 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
65exlimiv 1938 . . . . . . 7 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
7 eqid 2737 . . . . . . . 8 {𝑥} = {𝑥}
8 snex 5324 . . . . . . . . 9 {𝑥} ∈ V
9 eleq2 2826 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑦𝑢𝑦 ∈ {𝑥}))
10 eqeq1 2741 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
119, 10anbi12d 634 . . . . . . . . 9 (𝑢 = {𝑥} → ((𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥})))
128, 11spcev 3521 . . . . . . . 8 ((𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥}) → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
137, 12mpan2 691 . . . . . . 7 (𝑦 ∈ {𝑥} → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
146, 13impbii 212 . . . . . 6 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑦 ∈ {𝑥})
15 velsn 4557 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
16 equcom 2026 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
1714, 15, 163bitri 300 . . . . 5 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑥 = 𝑦)
1817rexbii 3170 . . . 4 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑥𝑋 𝑥 = 𝑦)
19 r19.42v 3263 . . . . . 6 (∃𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2019exbii 1855 . . . . 5 (∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
21 rexcom4 3172 . . . . 5 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}))
22 eluniab 4834 . . . . 5 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2320, 21, 223bitr4ri 307 . . . 4 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}))
24 risset 3186 . . . 4 (𝑦𝑋 ↔ ∃𝑥𝑋 𝑥 = 𝑦)
2518, 23, 243bitr4i 306 . . 3 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ 𝑦𝑋)
2625eqriv 2734 . 2 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = 𝑋
272, 26eqtr2i 2766 1 𝑋 = 𝐶
Colors of variables: wff setvar class
Syntax hints:  wa 399   = wceq 1543  wex 1787  wcel 2110  {cab 2714  wrex 3062  {csn 4541   cuni 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-rex 3067  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-sn 4542  df-pr 4544  df-uni 4820
This theorem is referenced by:  dissnref  22425  dissnlocfin  22426
  Copyright terms: Public domain W3C validator