Theorem pp0ex 4277

Description: {∅, {∅}} (the ordinal 2) is a set. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
pp0ex	⊢ {∅, {∅}} ∈ V

Proof of Theorem pp0ex

Step	Hyp	Ref	Expression
1		p0ex 4276	. . 3 ⊢ {∅} ∈ V
2	1	pwex 4271	. 2 ⊢ 𝒫 {∅} ∈ V
3		pwpw0ss 3886	. 2 ⊢ {∅, {∅}} ⊆ 𝒫 {∅}
4	2, 3	ssexi 4225	1 ⊢ {∅, {∅}} ∈ V

Syntax hints:   ∈ wcel 2200  Vcvv 2800  ∅c0 3492  𝒫 cpw 3650  {csn 3667  {cpr 3668

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-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674

This theorem is referenced by:  ord3ex  4278  ontr2exmid  4621  ordtri2or2exmidlem  4622  onsucelsucexmidlem  4625  regexmid  4631  reg2exmid  4632  reg3exmid  4676  nnregexmid  4717  acexmidlemcase  6008  acexmidlemv  6011  exmidpw2en  7097  exmidaclem  7413