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

Theorem uniintsn 4990
Description: Two ways to express "𝐴 is a singleton". See also en1 9023, en1b 9025, card1 9965, and eusn 4733. (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 4337 . . . . . 6 V ≠ ∅
2 inteq 4952 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 4965 . . . . . . . . . . 11 ∅ = V
42, 3eqtrdi 2786 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = V)
54adantl 480 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = V)
6 unieq 4918 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
7 uni0 4938 . . . . . . . . . . . 12 ∅ = ∅
86, 7eqtrdi 2786 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
9 eqeq1 2734 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ( 𝐴 = ∅ ↔ 𝐴 = ∅))
108, 9imbitrid 243 . . . . . . . . . 10 ( 𝐴 = 𝐴 → (𝐴 = ∅ → 𝐴 = ∅))
1110imp 405 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = ∅)
125, 11eqtr3d 2772 . . . . . . . 8 (( 𝐴 = 𝐴𝐴 = ∅) → V = ∅)
1312ex 411 . . . . . . 7 ( 𝐴 = 𝐴 → (𝐴 = ∅ → V = ∅))
1413necon3d 2959 . . . . . 6 ( 𝐴 = 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
151, 14mpi 20 . . . . 5 ( 𝐴 = 𝐴𝐴 ≠ ∅)
16 n0 4345 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
1715, 16sylib 217 . . . 4 ( 𝐴 = 𝐴 → ∃𝑥 𝑥𝐴)
18 vex 3476 . . . . . . 7 𝑥 ∈ V
19 vex 3476 . . . . . . 7 𝑦 ∈ V
2018, 19prss 4822 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21 uniss 4915 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐴 {𝑥, 𝑦} ⊆ 𝐴)
2221adantl 480 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
23 simpl 481 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 = 𝐴)
2422, 23sseqtrd 4021 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
25 intss 4972 . . . . . . . . . . 11 ({𝑥, 𝑦} ⊆ 𝐴 𝐴 {𝑥, 𝑦})
2625adantl 480 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 {𝑥, 𝑦})
2724, 26sstrd 3991 . . . . . . . . 9 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ {𝑥, 𝑦})
2818, 19unipr 4925 . . . . . . . . 9 {𝑥, 𝑦} = (𝑥𝑦)
2918, 19intpr 4985 . . . . . . . . 9 {𝑥, 𝑦} = (𝑥𝑦)
3027, 28, 293sstr3g 4025 . . . . . . . 8 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥𝑦) ⊆ (𝑥𝑦))
31 inss1 4227 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
32 ssun1 4171 . . . . . . . . 9 𝑥 ⊆ (𝑥𝑦)
3331, 32sstri 3990 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
34 eqss 3996 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
35 uneqin 4277 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ 𝑥 = 𝑦)
3634, 35bitr3i 276 . . . . . . . 8 (((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) ↔ 𝑥 = 𝑦)
3730, 33, 36sylanblc 587 . . . . . . 7 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
3837ex 411 . . . . . 6 ( 𝐴 = 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴𝑥 = 𝑦))
3920, 38biimtrid 241 . . . . 5 ( 𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4039alrimivv 1929 . . . 4 ( 𝐴 = 𝐴 → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4117, 40jca 510 . . 3 ( 𝐴 = 𝐴 → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
42 euabsn 4729 . . . 4 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
43 eleq1w 2814 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4443eu4 2609 . . . 4 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
45 abid2 2869 . . . . . 6 {𝑥𝑥𝐴} = 𝐴
4645eqeq1i 2735 . . . . 5 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4746exbii 1848 . . . 4 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
4842, 44, 473bitr3i 300 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
4941, 48sylib 217 . 2 ( 𝐴 = 𝐴 → ∃𝑥 𝐴 = {𝑥})
50 unisnv 4930 . . . 4 {𝑥} = 𝑥
51 unieq 4918 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
52 inteq 4952 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
5318intsn 4989 . . . . 5 {𝑥} = 𝑥
5452, 53eqtrdi 2786 . . . 4 (𝐴 = {𝑥} → 𝐴 = 𝑥)
5550, 51, 543eqtr4a 2796 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝐴)
5655exlimiv 1931 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
5749, 56impbii 208 1 ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 394  wal 1537   = wceq 1539  wex 1779  wcel 2104  ∃!weu 2560  {cab 2707  wne 2938  Vcvv 3472  cun 3945  cin 3946  wss 3947  c0 4321  {csn 4627  {cpr 4629   cuni 4907   cint 4949
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-sn 4628  df-pr 4630  df-uni 4908  df-int 4950
This theorem is referenced by:  uniintab  4991
  Copyright terms: Public domain W3C validator