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Theorem uniintsn 4989
Description: Two ways to express "𝐴 is a singleton". See also en1 9062, en1b 9063, card1 10005, and eusn 4734. (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 4350 . . . . . 6 V ≠ ∅
2 inteq 4953 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 4966 . . . . . . . . . . 11 ∅ = V
42, 3eqtrdi 2790 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = V)
54adantl 481 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = V)
6 unieq 4922 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
7 uni0 4939 . . . . . . . . . . . 12 ∅ = ∅
86, 7eqtrdi 2790 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
9 eqeq1 2738 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ( 𝐴 = ∅ ↔ 𝐴 = ∅))
108, 9imbitrid 244 . . . . . . . . . 10 ( 𝐴 = 𝐴 → (𝐴 = ∅ → 𝐴 = ∅))
1110imp 406 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = ∅)
125, 11eqtr3d 2776 . . . . . . . 8 (( 𝐴 = 𝐴𝐴 = ∅) → V = ∅)
1312ex 412 . . . . . . 7 ( 𝐴 = 𝐴 → (𝐴 = ∅ → V = ∅))
1413necon3d 2958 . . . . . 6 ( 𝐴 = 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
151, 14mpi 20 . . . . 5 ( 𝐴 = 𝐴𝐴 ≠ ∅)
16 n0 4358 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
1715, 16sylib 218 . . . 4 ( 𝐴 = 𝐴 → ∃𝑥 𝑥𝐴)
18 vex 3481 . . . . . . 7 𝑥 ∈ V
19 vex 3481 . . . . . . 7 𝑦 ∈ V
2018, 19prss 4824 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21 uniss 4919 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐴 {𝑥, 𝑦} ⊆ 𝐴)
2221adantl 481 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
23 simpl 482 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 = 𝐴)
2422, 23sseqtrd 4035 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
25 intss 4973 . . . . . . . . . . 11 ({𝑥, 𝑦} ⊆ 𝐴 𝐴 {𝑥, 𝑦})
2625adantl 481 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 {𝑥, 𝑦})
2724, 26sstrd 4005 . . . . . . . . 9 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ {𝑥, 𝑦})
2818, 19unipr 4928 . . . . . . . . 9 {𝑥, 𝑦} = (𝑥𝑦)
2918, 19intpr 4986 . . . . . . . . 9 {𝑥, 𝑦} = (𝑥𝑦)
3027, 28, 293sstr3g 4039 . . . . . . . 8 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥𝑦) ⊆ (𝑥𝑦))
31 inss1 4244 . . . . . . . . 9 (𝑥𝑦) ⊆ 𝑥
32 ssun1 4187 . . . . . . . . 9 𝑥 ⊆ (𝑥𝑦)
3331, 32sstri 4004 . . . . . . . 8 (𝑥𝑦) ⊆ (𝑥𝑦)
34 eqss 4010 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
35 uneqin 4294 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ 𝑥 = 𝑦)
3634, 35bitr3i 277 . . . . . . . 8 (((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) ↔ 𝑥 = 𝑦)
3730, 33, 36sylanblc 589 . . . . . . 7 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
3837ex 412 . . . . . 6 ( 𝐴 = 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴𝑥 = 𝑦))
3920, 38biimtrid 242 . . . . 5 ( 𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4039alrimivv 1925 . . . 4 ( 𝐴 = 𝐴 → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4117, 40jca 511 . . 3 ( 𝐴 = 𝐴 → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
42 euabsn 4730 . . . 4 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
43 eleq1w 2821 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4443eu4 2612 . . . 4 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
45 abid2 2876 . . . . . 6 {𝑥𝑥𝐴} = 𝐴
4645eqeq1i 2739 . . . . 5 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4746exbii 1844 . . . 4 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
4842, 44, 473bitr3i 301 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
4941, 48sylib 218 . 2 ( 𝐴 = 𝐴 → ∃𝑥 𝐴 = {𝑥})
50 unisnv 4931 . . . 4 {𝑥} = 𝑥
51 unieq 4922 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
52 inteq 4953 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
5318intsn 4988 . . . . 5 {𝑥} = 𝑥
5452, 53eqtrdi 2790 . . . 4 (𝐴 = {𝑥} → 𝐴 = 𝑥)
5550, 51, 543eqtr4a 2800 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝐴)
5655exlimiv 1927 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
5749, 56impbii 209 1 ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1534   = wceq 1536  wex 1775  wcel 2105  ∃!weu 2565  {cab 2711  wne 2937  Vcvv 3477  cun 3960  cin 3961  wss 3962  c0 4338  {csn 4630  {cpr 4632   cuni 4911   cint 4950
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-sn 4631  df-pr 4633  df-uni 4912  df-int 4951
This theorem is referenced by:  uniintab  4990
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