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Theorem uniintsn 4546
Description: Two ways to express "𝐴 is a singleton." See also en1 8064, en1b 8065, card1 8832, and eusn 4297. (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 3957 . . . . . 6 V ≠ ∅
2 inteq 4510 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 4522 . . . . . . . . . . 11 ∅ = V
42, 3syl6eq 2701 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = V)
54adantl 481 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = V)
6 unieq 4476 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
7 uni0 4497 . . . . . . . . . . . 12 ∅ = ∅
86, 7syl6eq 2701 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
9 eqeq1 2655 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ( 𝐴 = ∅ ↔ 𝐴 = ∅))
108, 9syl5ib 234 . . . . . . . . . 10 ( 𝐴 = 𝐴 → (𝐴 = ∅ → 𝐴 = ∅))
1110imp 444 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = ∅)
125, 11eqtr3d 2687 . . . . . . . 8 (( 𝐴 = 𝐴𝐴 = ∅) → V = ∅)
1312ex 449 . . . . . . 7 ( 𝐴 = 𝐴 → (𝐴 = ∅ → V = ∅))
1413necon3d 2844 . . . . . 6 ( 𝐴 = 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
151, 14mpi 20 . . . . 5 ( 𝐴 = 𝐴𝐴 ≠ ∅)
16 n0 3964 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
1715, 16sylib 208 . . . 4 ( 𝐴 = 𝐴 → ∃𝑥 𝑥𝐴)
18 vex 3234 . . . . . . 7 𝑥 ∈ V
19 vex 3234 . . . . . . 7 𝑦 ∈ V
2018, 19prss 4383 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21 uniss 4490 . . . . . . . . . . . . 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 3674 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
25 intss 4530 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐴 𝐴 {𝑥, 𝑦})
2625adantl 481 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 {𝑥, 𝑦})
2724, 26sstrd 3646 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ {𝑥, 𝑦})
2818, 19unipr 4481 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
2918, 19intpr 4542 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
3027, 28, 293sstr3g 3678 . . . . . . . . 9 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥𝑦) ⊆ (𝑥𝑦))
31 inss1 3866 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑥
32 ssun1 3809 . . . . . . . . . 10 𝑥 ⊆ (𝑥𝑦)
3331, 32sstri 3645 . . . . . . . . 9 (𝑥𝑦) ⊆ (𝑥𝑦)
3430, 33jctir 560 . . . . . . . 8 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
35 eqss 3651 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
36 uneqin 3911 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ 𝑥 = 𝑦)
3735, 36bitr3i 266 . . . . . . . 8 (((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) ↔ 𝑥 = 𝑦)
3834, 37sylib 208 . . . . . . 7 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
3938ex 449 . . . . . 6 ( 𝐴 = 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴𝑥 = 𝑦))
4020, 39syl5bi 232 . . . . 5 ( 𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4140alrimivv 1896 . . . 4 ( 𝐴 = 𝐴 → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4217, 41jca 553 . . 3 ( 𝐴 = 𝐴 → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
43 euabsn 4293 . . . 4 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
44 eleq1 2718 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4544eu4 2547 . . . 4 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
46 abid2 2774 . . . . . 6 {𝑥𝑥𝐴} = 𝐴
4746eqeq1i 2656 . . . . 5 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4847exbii 1814 . . . 4 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
4943, 45, 483bitr3i 290 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
5042, 49sylib 208 . 2 ( 𝐴 = 𝐴 → ∃𝑥 𝐴 = {𝑥})
5118unisn 4483 . . . 4 {𝑥} = 𝑥
52 unieq 4476 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
53 inteq 4510 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
5418intsn 4545 . . . . 5 {𝑥} = 𝑥
5553, 54syl6eq 2701 . . . 4 (𝐴 = {𝑥} → 𝐴 = 𝑥)
5651, 52, 553eqtr4a 2711 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝐴)
5756exlimiv 1898 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
5850, 57impbii 199 1 ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wal 1521   = wceq 1523  wex 1744  wcel 2030  ∃!weu 2498  {cab 2637  wne 2823  Vcvv 3231  cun 3605  cin 3606  wss 3607  c0 3948  {csn 4210  {cpr 4212   cuni 4468   cint 4507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-sn 4211  df-pr 4213  df-uni 4469  df-int 4508
This theorem is referenced by:  uniintab  4547
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