Theorem snelsingles 35886

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 3473	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3		eqcom 2742	. . . . . 6 ⊢ (𝑥 = 𝐴 ↔ 𝐴 = 𝑥)
4	3	exbii 1848	. . . . 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 4611	. . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥})
8	6, 7	eximii 1837	. 2 ⊢ ∃𝑥{𝐴} = {𝑥}
9		elsingles 35882	. 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
10	8, 9	mpbir 231	1 ⊢ {𝐴} ∈ Singletons

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459  {csn 4601   Singletons csingles 35803

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7727

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-symdif 4228  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-eprel 5553  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fo 6536  df-fv 6538  df-1st 7986  df-2nd 7987  df-txp 35818  df-singleton 35826  df-singles 35827

This theorem is referenced by: (None)