Theorem elsingles 36151

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 3453	. 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V)
2		vsnex 5371	. . . 4 ⊢ {𝑥} ∈ V
3		eleq1 2828	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
4	2, 3	mpbiri 259	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
5	4	exlimiv 1937	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6		eleq1 2828	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons ))
7		eqeq1 2744	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
8	7	exbidv 1928	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9		df-singles 36096	. . . . 5 ⊢ Singletons = ran Singleton
10	9	eleq2i 2832	. . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton)
11		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
12	11	elrn 5842	. . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13		vex 3436	. . . . . 6 ⊢ 𝑥 ∈ V
14	13, 11	brsingle 36150	. . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
15	14	exbii 1855	. . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
16	10, 12, 15	3bitri 298	. . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥})
17	6, 8, 16	vtoclbg 3505	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥}))
18	1, 5, 17	pm5.21nii 379	1 ⊢ (𝐴 ∈ Singletons ↔ ∃𝑥 𝐴 = {𝑥})

Syntax hints:   ↔ wb 207   = wceq 1547  ∃wex 1786   ∈ wcel 2119  Vcvv 3432  {csn 4562   class class class wbr 5079  ran crn 5626  Singletoncsingle 36071   Singletons csingles 36072

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2712  ax-sep 5225  ax-nul 5235  ax-pr 5369  ax-un 7685

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2719  df-cleq 2732  df-clel 2815  df-nfc 2889  df-ne 2936  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4188  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4846  df-br 5080  df-opab 5142  df-mpt 5161  df-id 5520  df-eprel 5525  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-fo 6498  df-fv 6500  df-1st 7938  df-2nd 7939  df-txp 36087  df-singleton 36095  df-singles 36096

This theorem is referenced by:  dfsingles2  36154  snelsingles  36155  funpartlem  36177