Theorem snex 3911

Description: A singleton whose element exists is a set. (Contributed by NM, 7-Aug-1994.) (Revised by Mario Carneiro, 24-May-2019.)

Hypothesis

Ref	Expression
snex.1	⊢ A ∈ V

Assertion

Ref	Expression
snex	⊢ {A} ∈ V

Proof of Theorem snex

Step	Hyp	Ref	Expression
1		snex.1	. 2 ⊢ A ∈ V
2		snexg 3910	. 2 ⊢ (A ∈ V → {A} ∈ V)
3	1, 2	ax-mp 7	1 ⊢ {A} ∈ V

Syntax hints:   ∈ wcel 1374  Vcvv 2535  {csn 3350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356

This theorem is referenced by: (None)