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Theorem elsingles 35919
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 3501 . 2 (𝐴 Singletons 𝐴 ∈ V)
2 vsnex 5434 . . . 4 {𝑥} ∈ V
3 eleq1 2829 . . . 4 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
42, 3mpbiri 258 . . 3 (𝐴 = {𝑥} → 𝐴 ∈ V)
54exlimiv 1930 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6 eleq1 2829 . . 3 (𝑦 = 𝐴 → (𝑦 Singletons 𝐴 Singletons ))
7 eqeq1 2741 . . . 4 (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
87exbidv 1921 . . 3 (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9 df-singles 35864 . . . . 5 Singletons = ran Singleton
109eleq2i 2833 . . . 4 (𝑦 Singletons 𝑦 ∈ ran Singleton)
11 vex 3484 . . . . 5 𝑦 ∈ V
1211elrn 5904 . . . 4 (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13 vex 3484 . . . . . 6 𝑥 ∈ V
1413, 11brsingle 35918 . . . . 5 (𝑥Singleton𝑦𝑦 = {𝑥})
1514exbii 1848 . . . 4 (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
1610, 12, 153bitri 297 . . 3 (𝑦 Singletons ↔ ∃𝑥 𝑦 = {𝑥})
176, 8, 16vtoclbg 3557 . 2 (𝐴 ∈ V → (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥}))
181, 5, 17pm5.21nii 378 1 (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 206   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  {csn 4626   class class class wbr 5143  ran crn 5686  Singletoncsingle 35839   Singletons csingles 35840
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-symdif 4253  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015  df-txp 35855  df-singleton 35863  df-singles 35864
This theorem is referenced by:  dfsingles2  35922  snelsingles  35923  funpartlem  35943
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