Theorem elsingles 35878

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 3484	. 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V)
2		vsnex 5414	. . . 4 ⊢ {𝑥} ∈ V
3		eleq1 2821	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
5	4	exlimiv 1929	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6		eleq1 2821	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons ))
7		eqeq1 2738	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
8	7	exbidv 1920	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9		df-singles 35823	. . . . 5 ⊢ Singletons = ran Singleton
10	9	eleq2i 2825	. . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton)
11		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
12	11	elrn 5884	. . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
14	13, 11	brsingle 35877	. . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
15	14	exbii 1847	. . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
16	10, 12, 15	3bitri 297	. . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥})
17	6, 8, 16	vtoclbg 3540	. 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 1539  ∃wex 1778   ∈ wcel 2107  Vcvv 3463  {csn 4606   class class class wbr 5123  ran crn 5666  Singletoncsingle 35798   Singletons csingles 35799

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412  ax-un 7737

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-symdif 4233  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-mpt 5206  df-id 5558  df-eprel 5564  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-fo 6547  df-fv 6549  df-1st 7996  df-2nd 7997  df-txp 35814  df-singleton 35822  df-singles 35823

This theorem is referenced by:  dfsingles2  35881  snelsingles  35882  funpartlem  35902