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Theorem uniintsn 4449
 Description: Two ways to express "𝐴 is a singleton." See also en1 7909, en1b 7910, card1 8677, and eusn 4209. (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 3883 . . . . . 6 V ≠ ∅
2 inteq 4413 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 4425 . . . . . . . . . . 11 ∅ = V
42, 3syl6eq 2660 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = V)
54adantl 481 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = V)
6 unieq 4380 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
7 uni0 4401 . . . . . . . . . . . 12 ∅ = ∅
86, 7syl6eq 2660 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
9 eqeq1 2614 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ( 𝐴 = ∅ ↔ 𝐴 = ∅))
108, 9syl5ib 233 . . . . . . . . . 10 ( 𝐴 = 𝐴 → (𝐴 = ∅ → 𝐴 = ∅))
1110imp 444 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = ∅)
125, 11eqtr3d 2646 . . . . . . . 8 (( 𝐴 = 𝐴𝐴 = ∅) → V = ∅)
1312ex 449 . . . . . . 7 ( 𝐴 = 𝐴 → (𝐴 = ∅ → V = ∅))
1413necon3d 2803 . . . . . 6 ( 𝐴 = 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
151, 14mpi 20 . . . . 5 ( 𝐴 = 𝐴𝐴 ≠ ∅)
16 n0 3890 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
1715, 16sylib 207 . . . 4 ( 𝐴 = 𝐴 → ∃𝑥 𝑥𝐴)
18 vex 3176 . . . . . . 7 𝑥 ∈ V
19 vex 3176 . . . . . . 7 𝑦 ∈ V
2018, 19prss 4291 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21 uniss 4394 . . . . . . . . . . . . 13 ({𝑥, 𝑦} ⊆ 𝐴 {𝑥, 𝑦} ⊆ 𝐴)
2221adantl 481 . . . . . . . . . . . 12 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
23 simpl 472 . . . . . . . . . . . 12 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 = 𝐴)
2422, 23sseqtrd 3604 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
25 intss 4433 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐴 𝐴 {𝑥, 𝑦})
2625adantl 481 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 {𝑥, 𝑦})
2724, 26sstrd 3578 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ {𝑥, 𝑦})
2818, 19unipr 4385 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
2918, 19intpr 4445 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
3027, 28, 293sstr3g 3608 . . . . . . . . 9 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥𝑦) ⊆ (𝑥𝑦))
31 inss1 3795 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑥
32 ssun1 3738 . . . . . . . . . 10 𝑥 ⊆ (𝑥𝑦)
3331, 32sstri 3577 . . . . . . . . 9 (𝑥𝑦) ⊆ (𝑥𝑦)
3430, 33jctir 559 . . . . . . . 8 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
35 eqss 3583 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
36 uneqin 3837 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ 𝑥 = 𝑦)
3735, 36bitr3i 265 . . . . . . . 8 (((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) ↔ 𝑥 = 𝑦)
3834, 37sylib 207 . . . . . . 7 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
3938ex 449 . . . . . 6 ( 𝐴 = 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴𝑥 = 𝑦))
4020, 39syl5bi 231 . . . . 5 ( 𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4140alrimivv 1843 . . . 4 ( 𝐴 = 𝐴 → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4217, 41jca 553 . . 3 ( 𝐴 = 𝐴 → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
43 euabsn 4205 . . . 4 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
44 eleq1 2676 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4544eu4 2506 . . . 4 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
46 abid2 2732 . . . . . 6 {𝑥𝑥𝐴} = 𝐴
4746eqeq1i 2615 . . . . 5 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4847exbii 1764 . . . 4 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
4943, 45, 483bitr3i 289 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
5042, 49sylib 207 . 2 ( 𝐴 = 𝐴 → ∃𝑥 𝐴 = {𝑥})
5118unisn 4387 . . . 4 {𝑥} = 𝑥
52 unieq 4380 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
53 inteq 4413 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
5418intsn 4448 . . . . 5 {𝑥} = 𝑥
5553, 54syl6eq 2660 . . . 4 (𝐴 = {𝑥} → 𝐴 = 𝑥)
5651, 52, 553eqtr4a 2670 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝐴)
5756exlimiv 1845 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
5850, 57impbii 198 1 ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  ∃!weu 2458  {cab 2596   ≠ wne 2780  Vcvv 3173   ∪ cun 3538   ∩ cin 3539   ⊆ wss 3540  ∅c0 3874  {csn 4125  {cpr 4127  ∪ cuni 4372  ∩ cint 4410 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-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-sn 4126  df-pr 4128  df-uni 4373  df-int 4411 This theorem is referenced by:  uniintab  4450
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