Theorem p0ex 4273

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 4268	. 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 4259

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  4274  undifexmid  4278  exmidexmid  4281  exmidundif  4291  exmidundifim  4292  exmid1stab  4293  ordtriexmidlem  4612  ontr2exmid  4618  onsucsssucexmid  4620  onsucelsucexmid  4623  regexmidlemm  4625  ordsoexmid  4655  ordtri2or2exmid  4664  ontri2orexmidim  4665  opthprc  4772  acexmidlema  6001  acexmidlem2  6007  tposexg  6415  2dom  6971  map1  6978  endisj  6996  ssfiexmid  7051  domfiexmid  7053  exmidpw  7086  exmidpw2en  7090  djuex  7226  exmidomni  7325  exmidonfinlem  7387  exmidfodomrlemr  7396  exmidfodomrlemrALT  7397  exmidaclem  7406  pw1dom2  7428  pw1ne1  7430  sbthom  16508