Theorem elsingles 36119

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 3451	. 2 ⊢ (𝐴 ∈ Singletons → 𝐴 ∈ V)
2		vsnex 5370	. . . 4 ⊢ {𝑥} ∈ V
3		eleq1 2825	. . . 4 ⊢ (𝐴 = {𝑥} → (𝐴 ∈ V ↔ {𝑥} ∈ V))
4	2, 3	mpbiri 258	. . 3 ⊢ (𝐴 = {𝑥} → 𝐴 ∈ V)
5	4	exlimiv 1932	. 2 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 ∈ V)
6		eleq1 2825	. . 3 ⊢ (𝑦 = 𝐴 → (𝑦 ∈ Singletons ↔ 𝐴 ∈ Singletons ))
7		eqeq1 2741	. . . 4 ⊢ (𝑦 = 𝐴 → (𝑦 = {𝑥} ↔ 𝐴 = {𝑥}))
8	7	exbidv 1923	. . 3 ⊢ (𝑦 = 𝐴 → (∃𝑥 𝑦 = {𝑥} ↔ ∃𝑥 𝐴 = {𝑥}))
9		df-singles 36064	. . . . 5 ⊢ Singletons = ran Singleton
10	9	eleq2i 2829	. . . 4 ⊢ (𝑦 ∈ Singletons ↔ 𝑦 ∈ ran Singleton)
11		vex 3434	. . . . 5 ⊢ 𝑦 ∈ V
12	11	elrn 5840	. . . 4 ⊢ (𝑦 ∈ ran Singleton ↔ ∃𝑥 𝑥Singleton𝑦)
13		vex 3434	. . . . . 6 ⊢ 𝑥 ∈ V
14	13, 11	brsingle 36118	. . . . 5 ⊢ (𝑥Singleton𝑦 ↔ 𝑦 = {𝑥})
15	14	exbii 1850	. . . 4 ⊢ (∃𝑥 𝑥Singleton𝑦 ↔ ∃𝑥 𝑦 = {𝑥})
16	10, 12, 15	3bitri 297	. . 3 ⊢ (𝑦 ∈ Singletons ↔ ∃𝑥 𝑦 = {𝑥})
17	6, 8, 16	vtoclbg 3503	. 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 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3430  {csn 4568   class class class wbr 5086  ran crn 5623  Singletoncsingle 36039   Singletons csingles 36040

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-symdif 4194  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-1st 7933  df-2nd 7934  df-txp 36055  df-singleton 36063  df-singles 36064

This theorem is referenced by:  dfsingles2  36122  snelsingles  36123  funpartlem  36145