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

Theorem uniintsn 4875
Description: Two ways to express "𝐴 is a singleton." See also en1 8559, en1b 8560, card1 9381, and eusn 4626. (Contributed by NM, 2-Aug-2010.)
Assertion
Ref Expression
uniintsn ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem uniintsn
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 vn0 4254 . . . . . 6 V ≠ ∅
2 inteq 4841 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 4852 . . . . . . . . . . 11 ∅ = V
42, 3eqtrdi 2849 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = V)
54adantl 485 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = V)
6 unieq 4811 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
7 uni0 4828 . . . . . . . . . . . 12 ∅ = ∅
86, 7eqtrdi 2849 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
9 eqeq1 2802 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ( 𝐴 = ∅ ↔ 𝐴 = ∅))
108, 9syl5ib 247 . . . . . . . . . 10 ( 𝐴 = 𝐴 → (𝐴 = ∅ → 𝐴 = ∅))
1110imp 410 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = ∅)
125, 11eqtr3d 2835 . . . . . . . 8 (( 𝐴 = 𝐴𝐴 = ∅) → V = ∅)
1312ex 416 . . . . . . 7 ( 𝐴 = 𝐴 → (𝐴 = ∅ → V = ∅))
1413necon3d 3008 . . . . . 6 ( 𝐴 = 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
151, 14mpi 20 . . . . 5 ( 𝐴 = 𝐴𝐴 ≠ ∅)
16 n0 4260 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
1715, 16sylib 221 . . . 4 ( 𝐴 = 𝐴 → ∃𝑥 𝑥𝐴)
18 vex 3444 . . . . . . 7 𝑥 ∈ V
19 vex 3444 . . . . . . 7 𝑦 ∈ V
2018, 19prss 4713 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21 uniss 4808 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐴 {𝑥, 𝑦} ⊆ 𝐴)
2221adantl 485 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
23 simpl 486 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 = 𝐴)
2422, 23sseqtrd 3955 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
25 intss 4859 . . . . . . . . . . 11 ({𝑥, 𝑦} ⊆ 𝐴 𝐴 {𝑥, 𝑦})
2625adantl 485 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 {𝑥, 𝑦})
2724, 26sstrd 3925 . . . . . . . . 9 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ {𝑥, 𝑦})
2818, 19unipr 4817 . . . . . . . . 9 {𝑥, 𝑦} = (𝑥𝑦)
2918, 19intpr 4871 . . . . . . . . 9 {𝑥, 𝑦} = (𝑥𝑦)
3027, 28, 293sstr3g 3959 . . . . . . . 8 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥𝑦) ⊆ (𝑥𝑦))
31 inss1 4155 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
32 ssun1 4099 . . . . . . . . 9 𝑥 ⊆ (𝑥𝑦)
3331, 32sstri 3924 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
34 eqss 3930 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
35 uneqin 4205 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ 𝑥 = 𝑦)
3634, 35bitr3i 280 . . . . . . . 8 (((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) ↔ 𝑥 = 𝑦)
3730, 33, 36sylanblc 592 . . . . . . 7 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
3837ex 416 . . . . . 6 ( 𝐴 = 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴𝑥 = 𝑦))
3920, 38syl5bi 245 . . . . 5 ( 𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4039alrimivv 1929 . . . 4 ( 𝐴 = 𝐴 → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4117, 40jca 515 . . 3 ( 𝐴 = 𝐴 → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
42 euabsn 4622 . . . 4 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
43 eleq1w 2872 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4443eu4 2676 . . . 4 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
45 abid2 2932 . . . . . 6 {𝑥𝑥𝐴} = 𝐴
4645eqeq1i 2803 . . . . 5 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4746exbii 1849 . . . 4 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
4842, 44, 473bitr3i 304 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
4941, 48sylib 221 . 2 ( 𝐴 = 𝐴 → ∃𝑥 𝐴 = {𝑥})
5018unisn 4820 . . . 4 {𝑥} = 𝑥
51 unieq 4811 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
52 inteq 4841 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
5318intsn 4874 . . . . 5 {𝑥} = 𝑥
5452, 53eqtrdi 2849 . . . 4 (𝐴 = {𝑥} → 𝐴 = 𝑥)
5550, 51, 543eqtr4a 2859 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝐴)
5655exlimiv 1931 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
5749, 56impbii 212 1 ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wal 1536   = wceq 1538  wex 1781  wcel 2111  ∃!weu 2628  {cab 2776  wne 2987  Vcvv 3441  cun 3879  cin 3880  wss 3881  c0 4243  {csn 4525  {cpr 4527   cuni 4800   cint 4838
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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-sn 4526  df-pr 4528  df-uni 4801  df-int 4839
This theorem is referenced by:  uniintab  4876
  Copyright terms: Public domain W3C validator