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Theorem ord3ex 4886
Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 6991. (Contributed by NM, 2-May-2009.)
Assertion
Ref Expression
ord3ex {∅, {∅}, {∅, {∅}}} ∈ V

Proof of Theorem ord3ex
StepHypRef Expression
1 df-tp 4215 . 2 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2 pwpr 4462 . . . 4 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3 pp0ex 4885 . . . . 5 {∅, {∅}} ∈ V
43pwex 4878 . . . 4 𝒫 {∅, {∅}} ∈ V
52, 4eqeltrri 2727 . . 3 ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6 snsspr2 4378 . . . 4 {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7 unss2 3817 . . . 4 ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
86, 7ax-mp 5 . . 3 ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
95, 8ssexi 4836 . 2 ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
101, 9eqeltri 2726 1 {∅, {∅}, {∅, {∅}}} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2030  Vcvv 3231  cun 3605  wss 3607  c0 3948  𝒫 cpw 4191  {csn 4210  {cpr 4212  {ctp 4214
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215
This theorem is referenced by: (None)
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