Theorem snelsingles 36095

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 3455	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3		eqcom 2744	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥)
4	3	exbii 1850	. . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥)
5	2, 4	bitri 275	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥)
6	1, 5	mpbi 230	. . 3 ⊢ ∃𝑥 𝐴 = 𝑥
7		sneq 4591	. . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥})
8	6, 7	eximii 1839	. 2 ⊢ ∃𝑥{𝐴} = {𝑥}
9		elsingles 36091	. 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
10	8, 9	mpbir 231	1 ⊢ {𝐴} ∈ Singletons

Syntax hints:   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3441  {csn 4581   Singletons csingles 36012

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 5242  ax-nul 5252  ax-pr 5378  ax-un 7682

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 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-symdif 4206  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-br 5100  df-opab 5162  df-mpt 5181  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 6449  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7935  df-2nd 7936  df-txp 36027  df-singleton 36035  df-singles 36036

This theorem is referenced by: (None)