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Theorem snex 4185
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 4184 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 5 1 {𝐴} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2148  Vcvv 2737  {csn 3592
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 4121  ax-pow 4174
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 2739  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598
This theorem is referenced by:  snelpw  4213  rext  4215  sspwb  4216  intid  4224  euabex  4225  mss  4226  exss  4227  opi1  4232  opeqsn  4252  opeqpr  4253  uniop  4255  snnex  4448  op1stb  4478  dtruex  4558  relop  4777  funopg  5250  fo1st  6157  fo2nd  6158  mapsn  6689  mapsnconst  6693  mapsncnv  6694  mapsnf1o2  6695  elixpsn  6734  ixpsnf1o  6735  ensn1  6795  mapsnen  6810  xpsnen  6820  endisj  6823  xpcomco  6825  xpassen  6829  phplem2  6852  findcard2  6888  findcard2s  6889  ac6sfi  6897  xpfi  6928  djuex  7041  0ct  7105  finomni  7137  exmidfodomrlemim  7199  djuassen  7215  cc2lem  7264  nn0ex  9181  fxnn0nninf  10437  inftonninf  10440  hashxp  10805  reldvg  14118
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