Theorem p0ex 4276

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 3818	. 2 ⊢ 𝒫 ∅ = {∅}
2		0ex 4214	. . 3 ⊢ ∅ ∈ V
3	2	pwex 4271	. 2 ⊢ 𝒫 ∅ ∈ V
4	1, 3	eqeltrri 2303	1 ⊢ {∅} ∈ V

Syntax hints:   ∈ wcel 2200  Vcvv 2800  ∅c0 3492  𝒫 cpw 3650  {csn 3667

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 4205  ax-nul 4213  ax-pow 4262

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 2802  df-dif 3200  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673

This theorem is referenced by:  pp0ex  4277  undifexmid  4281  exmidexmid  4284  exmidundif  4294  exmidundifim  4295  exmid1stab  4296  ordtriexmidlem  4615  ontr2exmid  4621  onsucsssucexmid  4623  onsucelsucexmid  4626  regexmidlemm  4628  ordsoexmid  4658  ordtri2or2exmid  4667  ontri2orexmidim  4668  opthprc  4775  acexmidlema  6004  acexmidlem2  6010  tposexg  6419  2dom  6975  map1  6982  endisj  7003  ssfiexmid  7058  domfiexmid  7060  exmidpw  7093  exmidpw2en  7097  djuex  7233  exmidomni  7332  exmidonfinlem  7394  exmidfodomrlemr  7403  exmidfodomrlemrALT  7404  exmidaclem  7413  pw1dom2  7435  pw1ne1  7437  sbthom  16566