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

Theorem unisngl 21140
 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 4381 . 2 𝐶 = {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}}
3 simpl 472 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦𝑢)
4 simpr 476 . . . . . . . . 9 ((𝑦𝑢𝑢 = {𝑥}) → 𝑢 = {𝑥})
53, 4eleqtrd 2690 . . . . . . . 8 ((𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
65exlimiv 1845 . . . . . . 7 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) → 𝑦 ∈ {𝑥})
7 eqid 2610 . . . . . . . 8 {𝑥} = {𝑥}
8 snex 4835 . . . . . . . . 9 {𝑥} ∈ V
9 eleq2 2677 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑦𝑢𝑦 ∈ {𝑥}))
10 eqeq1 2614 . . . . . . . . . 10 (𝑢 = {𝑥} → (𝑢 = {𝑥} ↔ {𝑥} = {𝑥}))
119, 10anbi12d 743 . . . . . . . . 9 (𝑢 = {𝑥} → ((𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥})))
128, 11spcev 3273 . . . . . . . 8 ((𝑦 ∈ {𝑥} ∧ {𝑥} = {𝑥}) → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
137, 12mpan2 703 . . . . . . 7 (𝑦 ∈ {𝑥} → ∃𝑢(𝑦𝑢𝑢 = {𝑥}))
146, 13impbii 198 . . . . . 6 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑦 ∈ {𝑥})
15 velsn 4141 . . . . . 6 (𝑦 ∈ {𝑥} ↔ 𝑦 = 𝑥)
16 equcom 1932 . . . . . 6 (𝑦 = 𝑥𝑥 = 𝑦)
1714, 15, 163bitri 285 . . . . 5 (∃𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ 𝑥 = 𝑦)
1817rexbii 3023 . . . 4 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑥𝑋 𝑥 = 𝑦)
19 r19.42v 3073 . . . . . 6 (∃𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ (𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2019exbii 1764 . . . . 5 (∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
21 rexcom4 3198 . . . . 5 (∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}) ↔ ∃𝑢𝑥𝑋 (𝑦𝑢𝑢 = {𝑥}))
22 eluniab 4383 . . . . 5 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑢(𝑦𝑢 ∧ ∃𝑥𝑋 𝑢 = {𝑥}))
2320, 21, 223bitr4ri 292 . . . 4 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ ∃𝑥𝑋𝑢(𝑦𝑢𝑢 = {𝑥}))
24 risset 3044 . . . 4 (𝑦𝑋 ↔ ∃𝑥𝑋 𝑥 = 𝑦)
2518, 23, 243bitr4i 291 . . 3 (𝑦 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} ↔ 𝑦𝑋)
2625eqriv 2607 . 2 {𝑢 ∣ ∃𝑥𝑋 𝑢 = {𝑥}} = 𝑋
272, 26eqtr2i 2633 1 𝑋 = 𝐶
 Colors of variables: wff setvar class Syntax hints:   ∧ wa 383   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∃wrex 2897  {csn 4125  ∪ cuni 4372 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373 This theorem is referenced by:  dissnref  21141  dissnlocfin  21142
 Copyright terms: Public domain W3C validator