Theorem elsingles 35571

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 3482	. 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V)
2		vsnex 5425	. . . 4 ⊢ {𝑥} ∈ V
3		eleq1 2813	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
4	2, 3	mpbiri 257	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
5	4	exlimiv 1925	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6		eleq1 2813	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons ))
7		eqeq1 2729	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
8	7	exbidv 1916	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9		df-singles 35516	. . . . 5 ⊢ Singletons = ran Singleton
10	9	eleq2i 2817	. . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton)
11		vex 3467	. . . . 5 ⊢ 𝑦 ∈ V
12	11	elrn 5890	. . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13		vex 3467	. . . . . 6 ⊢ 𝑥 ∈ V
14	13, 11	brsingle 35570	. . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
15	14	exbii 1842	. . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
16	10, 12, 15	3bitri 296	. . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥})
17	6, 8, 16	vtoclbg 3534	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}))
18	1, 5, 17	pm5.21nii 377	1 ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 205   = wceq 1533  ∃wex 1773   ∈ wcel 2098  Vcvv 3463  {csn 4624   class class class wbr 5143  ran crn 5673  Singletoncsingle 35491   Singletons csingles 35492

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423  ax-un 7738

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2703  df-cleq 2717  df-clel 2802  df-nfc 2877  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3942  df-un 3944  df-in 3946  df-ss 3956  df-symdif 4237  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-mpt 5227  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7991  df-2nd 7992  df-txp 35507  df-singleton 35515  df-singles 35516

This theorem is referenced by:  dfsingles2  35574  snelsingles  35575  funpartlem  35595