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Theorem elsingles 35571
Description: Membership in the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)
Assertion
Ref Expression
elsingles (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem elsingles
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 elex 3482 . 2 (𝐴 Singletons 𝐴 ∈ V)
2 vsnex 5425 . . . 4 {𝑥} ∈ V
3 eleq1 2813 . . . 4 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
42, 3mpbiri 257 . . 3 (𝐴 = {𝑥} → 𝐴 ∈ V)
54exlimiv 1925 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6 eleq1 2813 . . 3 (𝑦 = 𝐴 → (𝑦 Singletons 𝐴 Singletons ))
7 eqeq1 2729 . . . 4 (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
87exbidv 1916 . . 3 (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9 df-singles 35516 . . . . 5 Singletons = ran Singleton
109eleq2i 2817 . . . 4 (𝑦 Singletons 𝑦 ∈ ran Singleton)
11 vex 3467 . . . . 5 𝑦 ∈ V
1211elrn 5890 . . . 4 (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13 vex 3467 . . . . . 6 𝑥 ∈ V
1413, 11brsingle 35570 . . . . 5 (𝑥Singleton𝑦𝑦 = {𝑥})
1514exbii 1842 . . . 4 (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
1610, 12, 153bitri 296 . . 3 (𝑦 Singletons ↔ ∃𝑥 𝑦 = {𝑥})
176, 8, 16vtoclbg 3534 . 2 (𝐴 ∈ V → (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥}))
181, 5, 17pm5.21nii 377 1 (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 205   = wceq 1533  wex 1773  wcel 2098  Vcvv 3463  {csn 4624   class class class wbr 5143  ran crn 5673  Singletoncsingle 35491   Singletons csingles 35492
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-symdif 4237  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7991  df-2nd 7992  df-txp 35507  df-singleton 35515  df-singles 35516
This theorem is referenced by:  dfsingles2  35574  snelsingles  35575  funpartlem  35595
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