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Theorem snex 4186
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 4185 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 5 1 {𝐴} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2148  Vcvv 2738  {csn 3593
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-pow 4175
This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2740  df-in 3136  df-ss 3143  df-pw 3578  df-sn 3599
This theorem is referenced by:  snelpw  4214  rext  4216  sspwb  4217  intid  4225  euabex  4226  mss  4227  exss  4228  opi1  4233  opeqsn  4253  opeqpr  4254  uniop  4256  snnex  4449  op1stb  4479  dtruex  4559  relop  4778  funopg  5251  fo1st  6158  fo2nd  6159  mapsn  6690  mapsnconst  6694  mapsncnv  6695  mapsnf1o2  6696  elixpsn  6735  ixpsnf1o  6736  ensn1  6796  mapsnen  6811  xpsnen  6821  endisj  6824  xpcomco  6826  xpassen  6830  phplem2  6853  findcard2  6889  findcard2s  6890  ac6sfi  6898  xpfi  6929  djuex  7042  0ct  7106  finomni  7138  exmidfodomrlemim  7200  djuassen  7216  cc2lem  7265  nn0ex  9182  fxnn0nninf  10438  inftonninf  10441  hashxp  10806  reldvg  14151
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