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Theorem uniintsn 4443
Description: Two ways to express "𝐴 is a singleton." See also en1 7886, en1b 7887, card1 8654, and eusn 4208. (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 3882 . . . . . 6 V ≠ ∅
2 inteq 4407 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
3 int0 4419 . . . . . . . . . . 11 ∅ = V
42, 3syl6eq 2659 . . . . . . . . . 10 (𝐴 = ∅ → 𝐴 = V)
54adantl 480 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = V)
6 unieq 4374 . . . . . . . . . . . 12 (𝐴 = ∅ → 𝐴 = ∅)
7 uni0 4395 . . . . . . . . . . . 12 ∅ = ∅
86, 7syl6eq 2659 . . . . . . . . . . 11 (𝐴 = ∅ → 𝐴 = ∅)
9 eqeq1 2613 . . . . . . . . . . 11 ( 𝐴 = 𝐴 → ( 𝐴 = ∅ ↔ 𝐴 = ∅))
108, 9syl5ib 232 . . . . . . . . . 10 ( 𝐴 = 𝐴 → (𝐴 = ∅ → 𝐴 = ∅))
1110imp 443 . . . . . . . . 9 (( 𝐴 = 𝐴𝐴 = ∅) → 𝐴 = ∅)
125, 11eqtr3d 2645 . . . . . . . 8 (( 𝐴 = 𝐴𝐴 = ∅) → V = ∅)
1312ex 448 . . . . . . 7 ( 𝐴 = 𝐴 → (𝐴 = ∅ → V = ∅))
1413necon3d 2802 . . . . . 6 ( 𝐴 = 𝐴 → (V ≠ ∅ → 𝐴 ≠ ∅))
151, 14mpi 20 . . . . 5 ( 𝐴 = 𝐴𝐴 ≠ ∅)
16 n0 3889 . . . . 5 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
1715, 16sylib 206 . . . 4 ( 𝐴 = 𝐴 → ∃𝑥 𝑥𝐴)
18 vex 3175 . . . . . . 7 𝑥 ∈ V
19 vex 3175 . . . . . . 7 𝑦 ∈ V
2018, 19prss 4290 . . . . . 6 ((𝑥𝐴𝑦𝐴) ↔ {𝑥, 𝑦} ⊆ 𝐴)
21 uniss 4388 . . . . . . . . . . . . 13 ({𝑥, 𝑦} ⊆ 𝐴 {𝑥, 𝑦} ⊆ 𝐴)
2221adantl 480 . . . . . . . . . . . 12 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
23 simpl 471 . . . . . . . . . . . 12 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 = 𝐴)
2422, 23sseqtrd 3603 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ 𝐴)
25 intss 4427 . . . . . . . . . . . 12 ({𝑥, 𝑦} ⊆ 𝐴 𝐴 {𝑥, 𝑦})
2625adantl 480 . . . . . . . . . . 11 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝐴 {𝑥, 𝑦})
2724, 26sstrd 3577 . . . . . . . . . 10 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → {𝑥, 𝑦} ⊆ {𝑥, 𝑦})
2818, 19unipr 4379 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
2918, 19intpr 4439 . . . . . . . . . 10 {𝑥, 𝑦} = (𝑥𝑦)
3027, 28, 293sstr3g 3607 . . . . . . . . 9 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → (𝑥𝑦) ⊆ (𝑥𝑦))
31 inss1 3794 . . . . . . . . . 10 (𝑥𝑦) ⊆ 𝑥
32 ssun1 3737 . . . . . . . . . 10 𝑥 ⊆ (𝑥𝑦)
3331, 32sstri 3576 . . . . . . . . 9 (𝑥𝑦) ⊆ (𝑥𝑦)
3430, 33jctir 558 . . . . . . . 8 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
35 eqss 3582 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ ((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)))
36 uneqin 3836 . . . . . . . . 9 ((𝑥𝑦) = (𝑥𝑦) ↔ 𝑥 = 𝑦)
3735, 36bitr3i 264 . . . . . . . 8 (((𝑥𝑦) ⊆ (𝑥𝑦) ∧ (𝑥𝑦) ⊆ (𝑥𝑦)) ↔ 𝑥 = 𝑦)
3834, 37sylib 206 . . . . . . 7 (( 𝐴 = 𝐴 ∧ {𝑥, 𝑦} ⊆ 𝐴) → 𝑥 = 𝑦)
3938ex 448 . . . . . 6 ( 𝐴 = 𝐴 → ({𝑥, 𝑦} ⊆ 𝐴𝑥 = 𝑦))
4020, 39syl5bi 230 . . . . 5 ( 𝐴 = 𝐴 → ((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4140alrimivv 1842 . . . 4 ( 𝐴 = 𝐴 → ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦))
4217, 41jca 552 . . 3 ( 𝐴 = 𝐴 → (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
43 euabsn 4204 . . . 4 (∃!𝑥 𝑥𝐴 ↔ ∃𝑥{𝑥𝑥𝐴} = {𝑥})
44 eleq1 2675 . . . . 5 (𝑥 = 𝑦 → (𝑥𝐴𝑦𝐴))
4544eu4 2505 . . . 4 (∃!𝑥 𝑥𝐴 ↔ (∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)))
46 abid2 2731 . . . . . 6 {𝑥𝑥𝐴} = 𝐴
4746eqeq1i 2614 . . . . 5 ({𝑥𝑥𝐴} = {𝑥} ↔ 𝐴 = {𝑥})
4847exbii 1763 . . . 4 (∃𝑥{𝑥𝑥𝐴} = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥})
4943, 45, 483bitr3i 288 . . 3 ((∃𝑥 𝑥𝐴 ∧ ∀𝑥𝑦((𝑥𝐴𝑦𝐴) → 𝑥 = 𝑦)) ↔ ∃𝑥 𝐴 = {𝑥})
5042, 49sylib 206 . 2 ( 𝐴 = 𝐴 → ∃𝑥 𝐴 = {𝑥})
5118unisn 4381 . . . 4 {𝑥} = 𝑥
52 unieq 4374 . . . 4 (𝐴 = {𝑥} → 𝐴 = {𝑥})
53 inteq 4407 . . . . 5 (𝐴 = {𝑥} → 𝐴 = {𝑥})
5418intsn 4442 . . . . 5 {𝑥} = 𝑥
5553, 54syl6eq 2659 . . . 4 (𝐴 = {𝑥} → 𝐴 = 𝑥)
5651, 52, 553eqtr4a 2669 . . 3 (𝐴 = {𝑥} → 𝐴 = 𝐴)
5756exlimiv 1844 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 = 𝐴)
5850, 57impbii 197 1 ( 𝐴 = 𝐴 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382  wal 1472   = wceq 1474  wex 1694  wcel 1976  ∃!weu 2457  {cab 2595  wne 2779  Vcvv 3172  cun 3537  cin 3538  wss 3539  c0 3873  {csn 4124  {cpr 4126   cuni 4366   cint 4404
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-sn 4125  df-pr 4127  df-uni 4367  df-int 4405
This theorem is referenced by:  uniintab  4444
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