Theorem snelsingles 31199

Description: A singleton is a member of the class of all singletons. (Contributed by Scott Fenton, 19-Feb-2013.)

Hypothesis

Ref	Expression
snelsingles.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
snelsingles	⊢ {𝐴} ∈ Singletons

Proof of Theorem snelsingles

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snelsingles.1	. . . 4 ⊢ 𝐴 ∈ V
2		isset 3180	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3		eqcom 2617	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥)
4	3	exbii 1764	. . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥)
5	2, 4	bitri 263	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥)
6	1, 5	mpbi 219	. . 3 ⊢ ∃𝑥 𝐴 = 𝑥
7		sneq 4135	. . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥})
8	6, 7	eximii 1754	. 2 ⊢ ∃𝑥{𝐴} = {𝑥}
9		elsingles 31195	. 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
10	8, 9	mpbir 220	1 ⊢ {𝐴} ∈ Singletons

Syntax hints:   = wceq 1475  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  {csn 4125   Singletons csingles 31115

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-symdif 3806  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-eprel 4949  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060  df-txp 31130  df-singleton 31138  df-singles 31139

This theorem is referenced by: (None)