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Theorem elsingles 32469
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 3365 . 2 (𝐴 Singletons 𝐴 ∈ V)
2 snex 5064 . . . 4 {𝑥} ∈ V
3 eleq1 2832 . . . 4 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
42, 3mpbiri 249 . . 3 (𝐴 = {𝑥} → 𝐴 ∈ V)
54exlimiv 2025 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6 eleq1 2832 . . 3 (𝑦 = 𝐴 → (𝑦 Singletons 𝐴 Singletons ))
7 eqeq1 2769 . . . 4 (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
87exbidv 2016 . . 3 (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9 df-singles 32414 . . . . 5 Singletons = ran Singleton
109eleq2i 2836 . . . 4 (𝑦 Singletons 𝑦 ∈ ran Singleton)
11 vex 3353 . . . . 5 𝑦 ∈ V
1211elrn 5535 . . . 4 (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13 vex 3353 . . . . . 6 𝑥 ∈ V
1413, 11brsingle 32468 . . . . 5 (𝑥Singleton𝑦𝑦 = {𝑥})
1514exbii 1943 . . . 4 (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
1610, 12, 153bitri 288 . . 3 (𝑦 Singletons ↔ ∃𝑥 𝑦 = {𝑥})
176, 8, 16vtoclbg 3419 . 2 (𝐴 ∈ V → (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥}))
181, 5, 17pm5.21nii 369 1 (𝐴 Singletons ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 197   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  {csn 4334   class class class wbr 4809  ran crn 5278  Singletoncsingle 32389   Singletons csingles 32390
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-symdif 4005  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-eprel 5190  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367  df-txp 32405  df-singleton 32413  df-singles 32414
This theorem is referenced by:  dfsingles2  32472  snelsingles  32473  funpartlem  32493
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