Theorem p0ex 4278

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 3820	. 2 ⊢ 𝒫 ∅ = {∅}
2		0ex 4216	. . 3 ⊢ ∅ ∈ V
3	2	pwex 4273	. 2 ⊢ 𝒫 ∅ ∈ V
4	1, 3	eqeltrri 2305	1 ⊢ {∅} ∈ V

Syntax hints:   ∈ wcel 2202  Vcvv 2802  ∅c0 3494  𝒫 cpw 3652  {csn 3669

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-dif 3202  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675

This theorem is referenced by:  pp0ex  4279  undifexmid  4283  exmidexmid  4286  exmidundif  4296  exmidundifim  4297  exmid1stab  4298  ordtriexmidlem  4617  ontr2exmid  4623  onsucsssucexmid  4625  onsucelsucexmid  4628  regexmidlemm  4630  ordsoexmid  4660  ordtri2or2exmid  4669  ontri2orexmidim  4670  opthprc  4777  acexmidlema  6008  acexmidlem2  6014  tposexg  6423  2dom  6979  map1  6986  endisj  7007  ssfiexmid  7062  ssfiexmidt  7064  domfiexmid  7066  exmidpw  7099  exmidpw2en  7103  djuex  7241  exmidomni  7340  exmidonfinlem  7403  exmidfodomrlemr  7412  exmidfodomrlemrALT  7413  exmidaclem  7422  pw1dom2  7444  pw1ne1  7446  sbthom  16630