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Theorem uniintsn 4706
Description: Two ways to express "𝐴 is a singleton." See also en1 8259, en1b 8260, card1 9077, and eusn 4456. (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 4126 . . . . . 6 V ≠ ∅
2 inteq 4672 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 4683 . . . . . . . . . . 11 ∅ = V
42, 3syl6eq 2856 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = V)
54adantl 469 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = V)
6 unieq 4638 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
7 uni0 4659 . . . . . . . . . . . 12 ∅ = ∅
86, 7syl6eq 2856 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
9 eqeq1 2810 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ( 𝐴 = ∅ ↔ 𝐴 = ∅))
108, 9syl5ib 235 . . . . . . . . . 10 ( 𝐴 = 𝐴 → (𝐴 = ∅ → 𝐴 = ∅))
1110imp 395 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = ∅)
125, 11eqtr3d 2842 . . . . . . . 8 (( 𝐴 = 𝐴𝐴 = ∅) → V = ∅)
1312ex 399 . . . . . . 7 ( 𝐴 = 𝐴 → (𝐴 = ∅ → V = ∅))
1413necon3d 2999 . . . . . 6 ( 𝐴 = 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
151, 14mpi 20 . . . . 5 ( 𝐴 = 𝐴𝐴 ≠ ∅)
16 n0 4132 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
1715, 16sylib 209 . . . 4 ( 𝐴 = 𝐴 → ∃𝑥 𝑥𝐴)
18 vex 3394 . . . . . . 7 𝑥 ∈ V
19 vex 3394 . . . . . . 7 𝑦 ∈ V
2018, 19prss 4541 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21 uniss 4653 . . . . . . . . . . . . 13 ({𝑥, 𝑦} ⊆ 𝐴 {𝑥, 𝑦} ⊆ 𝐴)
2221adantl 469 . . . . . . . . . . . 12 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
23 simpl 470 . . . . . . . . . . . 12 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 = 𝐴)
2422, 23sseqtrd 3838 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
25 intss 4690 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐴 𝐴 {𝑥, 𝑦})
2625adantl 469 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 {𝑥, 𝑦})
2724, 26sstrd 3808 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ {𝑥, 𝑦})
2818, 19unipr 4643 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
2918, 19intpr 4702 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
3027, 28, 293sstr3g 3842 . . . . . . . . 9 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥𝑦) ⊆ (𝑥𝑦))
31 inss1 4029 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑥
32 ssun1 3975 . . . . . . . . . 10 𝑥 ⊆ (𝑥𝑦)
3331, 32sstri 3807 . . . . . . . . 9 (𝑥𝑦) ⊆ (𝑥𝑦)
3430, 33jctir 512 . . . . . . . 8 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
35 eqss 3813 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
36 uneqin 4080 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ 𝑥 = 𝑦)
3735, 36bitr3i 268 . . . . . . . 8 (((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) ↔ 𝑥 = 𝑦)
3834, 37sylib 209 . . . . . . 7 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
3938ex 399 . . . . . 6 ( 𝐴 = 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴𝑥 = 𝑦))
4020, 39syl5bi 233 . . . . 5 ( 𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4140alrimivv 2019 . . . 4 ( 𝐴 = 𝐴 → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4217, 41jca 503 . . 3 ( 𝐴 = 𝐴 → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
43 euabsn 4452 . . . 4 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
44 eleq1w 2868 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4544eu4 2681 . . . 4 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
46 abid2 2929 . . . . . 6 {𝑥𝑥𝐴} = 𝐴
4746eqeq1i 2811 . . . . 5 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4847exbii 1933 . . . 4 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
4943, 45, 483bitr3i 292 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
5042, 49sylib 209 . 2 ( 𝐴 = 𝐴 → ∃𝑥 𝐴 = {𝑥})
5118unisn 4646 . . . 4 {𝑥} = 𝑥
52 unieq 4638 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
53 inteq 4672 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
5418intsn 4705 . . . . 5 {𝑥} = 𝑥
5553, 54syl6eq 2856 . . . 4 (𝐴 = {𝑥} → 𝐴 = 𝑥)
5651, 52, 553eqtr4a 2866 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝐴)
5756exlimiv 2021 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
5850, 57impbii 200 1 ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384  wal 1635   = wceq 1637  wex 1859  wcel 2156  ∃!weu 2630  {cab 2792  wne 2978  Vcvv 3391  cun 3767  cin 3768  wss 3769  c0 4116  {csn 4370  {cpr 4372   cuni 4630   cint 4669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-v 3393  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-sn 4371  df-pr 4373  df-uni 4631  df-int 4670
This theorem is referenced by:  uniintab  4707
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