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Theorem ord3ex 5387
Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 7755. (Contributed by NM, 2-May-2009.)
Assertion
Ref Expression
ord3ex {∅, {∅}, {∅, {∅}}} ∈ V

Proof of Theorem ord3ex
StepHypRef Expression
1 df-tp 4631 . 2 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2 pwpr 4901 . . . 4 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3 pp0ex 5386 . . . . 5 {∅, {∅}} ∈ V
43pwex 5380 . . . 4 𝒫 {∅, {∅}} ∈ V
52, 4eqeltrri 2838 . . 3 ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6 snsspr2 4815 . . . 4 {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7 unss2 4187 . . . 4 ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
86, 7ax-mp 5 . . 3 ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
95, 8ssexi 5322 . 2 ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
101, 9eqeltri 2837 1 {∅, {∅}, {∅, {∅}}} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 2108  Vcvv 3480  cun 3949  wss 3951  c0 4333  𝒫 cpw 4600  {csn 4626  {cpr 4628  {ctp 4630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pow 5365
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-pw 4602  df-sn 4627  df-pr 4629  df-tp 4631
This theorem is referenced by: (None)
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