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Theorem elsingles 32331
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 3352 . 2 (𝐴 Singletons 𝐴 ∈ V)
2 snex 5057 . . . 4 {𝑥} ∈ V
3 eleq1 2827 . . . 4 (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
42, 3mpbiri 248 . . 3 (𝐴 = {𝑥} → 𝐴 ∈ V)
54exlimiv 2007 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6 eleq1 2827 . . 3 (𝑦 = 𝐴 → (𝑦 Singletons 𝐴 Singletons ))
7 eqeq1 2764 . . . 4 (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
87exbidv 1999 . . 3 (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9 df-singles 32276 . . . . 5 Singletons = ran Singleton
109eleq2i 2831 . . . 4 (𝑦 Singletons 𝑦 ∈ ran Singleton)
11 vex 3343 . . . . 5 𝑦 ∈ V
1211elrn 5521 . . . 4 (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13 vex 3343 . . . . . 6 𝑥 ∈ V
1413, 11brsingle 32330 . . . . 5 (𝑥Singleton𝑦𝑦 = {𝑥})
1514exbii 1923 . . . 4 (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
1610, 12, 153bitri 286 . . 3 (𝑦 Singletons ↔ ∃𝑥 𝑦 = {𝑥})
176, 8, 16vtoclbg 3407 . 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 196   = wceq 1632  wex 1853  wcel 2139  Vcvv 3340  {csn 4321   class class class wbr 4804  ran crn 5267  Singletoncsingle 32251   Singletons csingles 32252
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7114
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-symdif 3987  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-eprel 5179  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-fo 6055  df-fv 6057  df-1st 7333  df-2nd 7334  df-txp 32267  df-singleton 32275  df-singles 32276
This theorem is referenced by:  dfsingles2  32334  snelsingles  32335  funpartlem  32355
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