Theorem elsingles 35900

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.

Step	Hyp	Ref	Expression
1		elex 3499	. 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V)
2		vsnex 5440	. . . 4 ⊢ {𝑥} ∈ V
3		eleq1 2827	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
5	4	exlimiv 1928	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6		eleq1 2827	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons ))
7		eqeq1 2739	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
8	7	exbidv 1919	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9		df-singles 35845	. . . . 5 ⊢ Singletons = ran Singleton
10	9	eleq2i 2831	. . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton)
11		vex 3482	. . . . 5 ⊢ 𝑦 ∈ V
12	11	elrn 5907	. . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13		vex 3482	. . . . . 6 ⊢ 𝑥 ∈ V
14	13, 11	brsingle 35899	. . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
15	14	exbii 1845	. . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
16	10, 12, 15	3bitri 297	. . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥})
17	6, 8, 16	vtoclbg 3557	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}))
18	1, 5, 17	pm5.21nii 378	1 ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 206   = wceq 1537  ∃wex 1776   ∈ wcel 2106  Vcvv 3478  {csn 4631   class class class wbr 5148  ran crn 5690  Singletoncsingle 35820   Singletons csingles 35821

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-symdif 4259  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-txp 35836  df-singleton 35844  df-singles 35845

This theorem is referenced by:  dfsingles2  35903  snelsingles  35904  funpartlem  35924