Theorem p0ex 4271

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

Step	Hyp	Ref	Expression
1		pw0 3814	. 2 ⊢ 𝒫 ∅ = {∅}
2		0ex 4210	. . 3 ⊢ ∅ ∈ V
3	2	pwex 4266	. 2 ⊢ 𝒫 ∅ ∈ V
4	1, 3	eqeltrri 2303	1 ⊢ {∅} ∈ V

Syntax hints:   ∈ wcel 2200  Vcvv 2799  ∅c0 3491  𝒫 cpw 3649  {csn 3666

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-dif 3199  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672

This theorem is referenced by:  pp0ex  4272  undifexmid  4276  exmidexmid  4279  exmidundif  4289  exmidundifim  4290  exmid1stab  4291  ordtriexmidlem  4610  ontr2exmid  4616  onsucsssucexmid  4618  onsucelsucexmid  4621  regexmidlemm  4623  ordsoexmid  4653  ordtri2or2exmid  4662  ontri2orexmidim  4663  opthprc  4769  acexmidlema  5991  acexmidlem2  5997  tposexg  6402  2dom  6956  map1  6963  endisj  6979  ssfiexmid  7034  domfiexmid  7036  exmidpw  7066  exmidpw2en  7070  djuex  7206  exmidomni  7305  exmidonfinlem  7367  exmidfodomrlemr  7376  exmidfodomrlemrALT  7377  exmidaclem  7386  pw1dom2  7408  pw1ne1  7410  sbthom  16353