Theorem snelsingles 35917

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 3464	. . . . 5 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴)
3		eqcom 2737	. . . . . 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 4602	. . 3 ⊢ (𝐴 = 𝑥 → {𝐴} = {𝑥})
8	6, 7	eximii 1837	. 2 ⊢ ∃𝑥{𝐴} = {𝑥}
9		elsingles 35913	. 2 ⊢ ({𝐴} ∈ Singletons ↔ ∃𝑥{𝐴} = {𝑥})
10	8, 9	mpbir 231	1 ⊢ {𝐴} ∈ Singletons

Syntax hints:   = wceq 1540  ∃wex 1779   ∈ wcel 2109  Vcvv 3450  {csn 4592   Singletons csingles 35834

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390  ax-un 7714

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 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-symdif 4219  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-eprel 5541  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-fo 6520  df-fv 6522  df-1st 7971  df-2nd 7972  df-txp 35849  df-singleton 35857  df-singles 35858

This theorem is referenced by: (None)