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Theorem elsingles 31195
 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 3185 . 2 (𝐴 Singletons 𝐴 ∈ V)
2 snex 4835 . . . 4 {𝑥} ∈ V
3 eleq1 2676 . . . 4 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
42, 3mpbiri 247 . . 3 (𝐴 = {𝑥} → 𝐴 ∈ V)
54exlimiv 1845 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6 eleq1 2676 . . 3 (𝑦 = 𝐴 → (𝑦 Singletons 𝐴 Singletons ))
7 eqeq1 2614 . . . 4 (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
87exbidv 1837 . . 3 (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9 df-singles 31139 . . . . 5 Singletons = ran Singleton
109eleq2i 2680 . . . 4 (𝑦 Singletons 𝑦 ∈ ran Singleton)
11 vex 3176 . . . . 5 𝑦 ∈ V
1211elrn 5287 . . . 4 (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13 vex 3176 . . . . . 6 𝑥 ∈ V
1413, 11brsingle 31194 . . . . 5 (𝑥Singleton𝑦𝑦 = {𝑥})
1514exbii 1764 . . . 4 (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
1610, 12, 153bitri 285 . . 3 (𝑦 Singletons ↔ ∃𝑥 𝑦 = {𝑥})
176, 8, 16vtoclbg 3240 . 2 (𝐴 ∈ V → (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥}))
181, 5, 17pm5.21nii 367 1 (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 195   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  {csn 4125   class class class wbr 4583  ran crn 5039  Singletoncsingle 31114   Singletons csingles 31115 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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 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-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138  df-singles 31139 This theorem is referenced by:  dfsingles2  31198  snelsingles  31199  funpartlem  31219
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