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Theorem df3o3 38997
Description: Ordinal 3 , fully expanded. (Contributed by RP, 8-Jul-2021.)
Assertion
Ref Expression
df3o3 3𝑜 = {∅, {∅}, {∅, {∅}}}

Proof of Theorem df3o3
StepHypRef Expression
1 df-3o 7766 . 2 3𝑜 = suc 2𝑜
2 df2o2 7779 . . . 4 2𝑜 = {∅, {∅}}
32sneqi 4345 . . . 4 {2𝑜} = {{∅, {∅}}}
42, 3uneq12i 3927 . . 3 (2𝑜 ∪ {2𝑜}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5 df-suc 5914 . . 3 suc 2𝑜 = (2𝑜 ∪ {2𝑜})
6 df-tp 4339 . . 3 {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
74, 5, 63eqtr4i 2797 . 2 suc 2𝑜 = {∅, {∅}, {∅, {∅}}}
81, 7eqtri 2787 1 3𝑜 = {∅, {∅}, {∅, {∅}}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cun 3730  c0 4079  {csn 4334  {cpr 4336  {ctp 4338  suc csuc 5910  2𝑜c2o 7758  3𝑜c3o 7759
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-dif 3735  df-un 3737  df-nul 4080  df-sn 4335  df-pr 4337  df-tp 4339  df-suc 5914  df-1o 7764  df-2o 7765  df-3o 7766
This theorem is referenced by: (None)
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