Theorem snelsingles 34203

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 3443	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3		eqcom 2746	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥)
4	3	exbii 1853	. . . . 5 ⊢ (∃𝑥 𝑥 = 𝐴 ↔ ∃𝑥 𝐴 = 𝑥)
5	2, 4	bitri 274	. . . 4 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝐴 = 𝑥)
6	1, 5	mpbi 229	. . 3 ⊢ ∃𝑥 𝐴 = 𝑥
7		sneq 4576	. . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥})
8	6, 7	eximii 1842	. 2 ⊢ ∃𝑥{𝐴} = {𝑥}
9		elsingles 34199	. 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
10	8, 9	mpbir 230	1 ⊢ {𝐴} ∈ Singletons

Syntax hints:   = wceq 1541  ∃wex 1785   ∈ wcel 2109  Vcvv 3430  {csn 4566   Singletons csingles 34120

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-10 2140  ax-11 2157  ax-12 2174  ax-ext 2710  ax-sep 5226  ax-nul 5233  ax-pr 5355  ax-un 7579

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-nf 1790  df-sb 2071  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-symdif 4181  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-mpt 5162  df-id 5488  df-eprel 5494  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-iota 6388  df-fun 6432  df-fn 6433  df-f 6434  df-fo 6436  df-fv 6438  df-1st 7817  df-2nd 7818  df-txp 34135  df-singleton 34143  df-singles 34144

This theorem is referenced by: (None)