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  6009  acexmidlem2  6015  tposexg  6424  2dom  6980  map1  6987  endisj  7008  ssfiexmid  7063  ssfiexmidt  7065  domfiexmid  7067  exmidpw  7100  exmidpw2en  7104  djuex  7242  exmidomni  7341  exmidonfinlem  7404  exmidfodomrlemr  7413  exmidfodomrlemrALT  7414  exmidaclem  7423  pw1dom2  7445  pw1ne1  7447  sbthom  16656