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Theorem p0ex 4206
Description: The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)
Assertion
Ref Expression
p0ex {∅} ∈ V

Proof of Theorem p0ex
StepHypRef Expression
1 pw0 3754 . 2 𝒫 ∅ = {∅}
2 0ex 4145 . . 3 ∅ ∈ V
32pwex 4201 . 2 𝒫 ∅ ∈ V
41, 3eqeltrri 2263 1 {∅} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2160  Vcvv 2752  c0 3437  𝒫 cpw 3590  {csn 3607
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-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2163  ax-ext 2171  ax-sep 4136  ax-nul 4144  ax-pow 4192
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-v 2754  df-dif 3146  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613
This theorem is referenced by:  pp0ex  4207  undifexmid  4211  exmidexmid  4214  exmidundif  4224  exmidundifim  4225  exmid1stab  4226  ordtriexmidlem  4536  ontr2exmid  4542  onsucsssucexmid  4544  onsucelsucexmid  4547  regexmidlemm  4549  ordsoexmid  4579  ordtri2or2exmid  4588  ontri2orexmidim  4589  opthprc  4695  acexmidlema  5887  acexmidlem2  5893  tposexg  6283  2dom  6831  map1  6838  endisj  6850  ssfiexmid  6904  domfiexmid  6906  exmidpw  6936  exmidpw2en  6940  djuex  7072  exmidomni  7170  exmidonfinlem  7222  exmidfodomrlemr  7231  exmidfodomrlemrALT  7232  exmidaclem  7237  pw1dom2  7256  pw1ne1  7258  sbthom  15236
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