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Theorem snex 3983
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
StepHypRef Expression
1 snex.1 . 2 𝐴 ∈ V
2 snexg 3982 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 7 1 {𝐴} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1434  Vcvv 2612  {csn 3422
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-v 2614  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428
This theorem is referenced by:  snelpw  4003  rext  4005  sspwb  4006  intid  4014  euabex  4015  mss  4016  exss  4017  opi1  4022  opeqsn  4042  opeqpr  4043  uniop  4045  snnex  4234  op1stb  4262  dtruex  4337  relop  4543  funopg  5000  fo1st  5862  fo2nd  5863  mapsn  6376  mapsnconst  6380  mapsncnv  6381  mapsnf1o2  6382  ensn1  6442  mapsnen  6457  xpsnen  6466  endisj  6469  xpcomco  6471  xpassen  6475  phplem2  6498  findcard2  6534  findcard2s  6535  ac6sfi  6543  xpfi  6564  djuex  6642  finomni  6700  exmidfodomrlemim  6729  nn0ex  8570  fxnn0nninf  9732  inftonninf  9735  hashxp  10068
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