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Theorem p0ex 3996
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 3567 . 2 𝒫 ∅ = {∅}
2 0ex 3940 . . 3 ∅ ∈ V
32pwex 3991 . 2 𝒫 ∅ ∈ V
41, 3eqeltrri 2158 1 {∅} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 1436  Vcvv 2615  c0 3275  𝒫 cpw 3415  {csn 3431
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-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-nul 3939  ax-pow 3983
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-in 2994  df-ss 3001  df-nul 3276  df-pw 3417  df-sn 3437
This theorem is referenced by:  pp0ex  3997  undifexmid  4001  exmidexmid  4004  exmidundif  4008  ordtriexmidlem  4308  ontr2exmid  4313  onsucsssucexmid  4315  onsucelsucexmid  4318  regexmidlemm  4320  ordsoexmid  4350  ordtri2or2exmid  4359  opthprc  4456  acexmidlema  5598  acexmidlem2  5604  tposexg  5971  2dom  6468  map1  6475  endisj  6486  ssfiexmid  6538  domfiexmid  6540  exmidpw  6570  djuex  6673  exmidomni  6735  exmidfodomrlemr  6765  exmidfodomrlemrALT  6766  pw1dom2  11319
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