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Theorem ord3ex 4219
Description: The ordinal number 3 is a set, proved without the Axiom of Union. (Contributed by NM, 2-May-2009.)
Assertion
Ref Expression
ord3ex {∅, {∅}, {∅, {∅}}} ∈ V

Proof of Theorem ord3ex
StepHypRef Expression
1 df-tp 3626 . 2 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
2 pp0ex 4218 . . . . 5 {∅, {∅}} ∈ V
32pwex 4212 . . . 4 𝒫 {∅, {∅}} ∈ V
4 pwprss 3831 . . . 4 ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ⊆ 𝒫 {∅, {∅}}
53, 4ssexi 4167 . . 3 ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}) ∈ V
6 snsspr2 3767 . . . 4 {{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}}
7 unss2 3330 . . . 4 ({{∅, {∅}}} ⊆ {{{∅}}, {∅, {∅}}} → ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}}))
86, 7ax-mp 5 . . 3 ({∅, {∅}} ∪ {{∅, {∅}}}) ⊆ ({∅, {∅}} ∪ {{{∅}}, {∅, {∅}}})
95, 8ssexi 4167 . 2 ({∅, {∅}} ∪ {{∅, {∅}}}) ∈ V
101, 9eqeltri 2266 1 {∅, {∅}, {∅, {∅}}} ∈ V
Colors of variables: wff set class
Syntax hints:  wcel 2164  Vcvv 2760  cun 3151  wss 3153  c0 3446  𝒫 cpw 3601  {csn 3618  {cpr 3619  {ctp 3620
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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-pr 3625  df-tp 3626
This theorem is referenced by: (None)
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