Theorem p0ex 4272

Description: The power set of the empty set (the ordinal 1) is a set. (Contributed by NM, 23-Dec-1993.)

Assertion

Ref	Expression
p0ex	|- { (/) } e. _V

Proof of Theorem p0ex

Step	Hyp	Ref	Expression
1		pw0 3815	. 2 |- ~P (/) = { (/) }
2		0ex 4211	. . 3 |- (/) e. _V
3	2	pwex 4267	. 2 |- ~P (/) e. _V
4	1, 3	eqeltrri 2303	1 |- { (/) } e. _V

Syntax hints:   e. wcel 2200   _Vcvv 2799   (/)c0 3491   ~Pcpw 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 4202  ax-nul 4210  ax-pow 4258

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  4273  undifexmid  4277  exmidexmid  4280  exmidundif  4290  exmidundifim  4291  exmid1stab  4292  ordtriexmidlem  4611  ontr2exmid  4617  onsucsssucexmid  4619  onsucelsucexmid  4622  regexmidlemm  4624  ordsoexmid  4654  ordtri2or2exmid  4663  ontri2orexmidim  4664  opthprc  4770  acexmidlema  5992  acexmidlem2  5998  tposexg  6404  2dom  6958  map1  6965  endisj  6983  ssfiexmid  7038  domfiexmid  7040  exmidpw  7070  exmidpw2en  7074  djuex  7210  exmidomni  7309  exmidonfinlem  7371  exmidfodomrlemr  7380  exmidfodomrlemrALT  7381  exmidaclem  7390  pw1dom2  7412  pw1ne1  7414  sbthom  16394