Theorem snex 3928

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 3927	. 2 ⊢ (A ∈ V → {A} ∈ V)
3	1, 2	ax-mp 7	1 ⊢ {A} ∈ V

Syntax hints:   ∈ wcel 1390  Vcvv 2551  {csn 3367

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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by:  ensn1  6212  xpsnen  6231  endisj  6234  xpcomco  6236  xpassen  6240  nn0ex  7963