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Theorem snex 3978
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 3977 . 2 (𝐴 ∈ V → {𝐴} ∈ V)
31, 2ax-mp 7 1 {𝐴} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1434  Vcvv 2611  {csn 3417
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 3917  ax-pow 3969
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 2613  df-in 2989  df-ss 2996  df-pw 3403  df-sn 3423
This theorem is referenced by:  snelpw  3998  rext  4000  sspwb  4001  intid  4009  euabex  4010  mss  4011  exss  4012  opi1  4017  opeqsn  4037  opeqpr  4038  uniop  4040  snnex  4229  op1stb  4257  dtruex  4332  relop  4537  funopg  4987  fo1st  5841  fo2nd  5842  ensn1  6371  xpsnen  6393  endisj  6396  xpcomco  6398  xpassen  6402  phplem2  6416  findcard2  6452  findcard2s  6453  ac6sfi  6461  xpfi  6479  djuex  6560  finomni  6590  nn0ex  8447  hashxp  9936
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