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Theorem ord3ex 4777
Description: The ordinal number 3 is a set, proved without the Axiom of Union ax-un 6824. (Contributed by NM, 2-May-2009.)
Assertion
Ref Expression
ord3ex {∅, {∅}, {∅, {∅}}} ∈ V

Proof of Theorem ord3ex
StepHypRef Expression
1 df-tp 4129 . 2 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2 pwpr 4362 . . . 4 𝒫 {∅, {∅}} = ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
3 pp0ex 4776 . . . . 5 {∅, {∅}} ∈ V
43pwex 4769 . . . 4 𝒫 {∅, {∅}} ∈ V
52, 4eqeltrri 2684 . . 3 ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6 snsspr2 4285 . . . 4 {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7 unss2 3745 . . . 4 ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
86, 7ax-mp 5 . . 3 ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
95, 8ssexi 4726 . 2 ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
101, 9eqeltri 2683 1 {∅, {∅}, {∅, {∅}}} ∈ V
Colors of variables: wff setvar class
Syntax hints:  wcel 1976  Vcvv 3172  cun 3537  wss 3539  c0 3873  𝒫 cpw 4107  {csn 4124  {cpr 4126  {ctp 4128
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-v 3174  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-pw 4109  df-sn 4125  df-pr 4127  df-tp 4129
This theorem is referenced by: (None)
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