Theorem p0ex 4284

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 3825	. 2 ⊢ 𝒫 ∅ = {∅}
2		0ex 4221	. . 3 ⊢ ∅ ∈ V
3	2	pwex 4279	. 2 ⊢ 𝒫 ∅ ∈ V
4	1, 3	eqeltrri 2305	1 ⊢ {∅} ∈ V

Syntax hints:   ∈ wcel 2202  Vcvv 2803  ∅c0 3496  𝒫 cpw 3656  {csn 3673

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-v 2805  df-dif 3203  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679

This theorem is referenced by:  pp0ex  4285  undifexmid  4289  exmidexmid  4292  exmidundif  4302  exmidundifim  4303  exmid1stab  4304  ordtriexmidlem  4623  ontr2exmid  4629  onsucsssucexmid  4631  onsucelsucexmid  4634  regexmidlemm  4636  ordsoexmid  4666  ordtri2or2exmid  4675  ontri2orexmidim  4676  opthprc  4783  acexmidlema  6019  acexmidlem2  6025  tposexg  6467  2dom  7023  map1  7030  endisj  7051  ssfiexmid  7106  ssfiexmidt  7108  domfiexmid  7110  exmidpw  7143  exmidpw2en  7147  djuex  7285  exmidomni  7384  exmidonfinlem  7447  exmidfodomrlemr  7456  exmidfodomrlemrALT  7457  exmidaclem  7466  pw1dom2  7488  pw1ne1  7490  sbthom  16737