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Theorem df2o3 6151
Description: Expanded value of the ordinal number 2. (Contributed by Mario Carneiro, 14-Aug-2015.)
Assertion
Ref Expression
df2o3 2𝑜 = {∅, 1𝑜}

Proof of Theorem df2o3
StepHypRef Expression
1 df-2o 6138 . 2 2𝑜 = suc 1𝑜
2 df-suc 4174 . 2 suc 1𝑜 = (1𝑜 ∪ {1𝑜})
3 df1o2 6150 . . . 4 1𝑜 = {∅}
43uneq1i 3139 . . 3 (1𝑜 ∪ {1𝑜}) = ({∅} ∪ {1𝑜})
5 df-pr 3438 . . 3 {∅, 1𝑜} = ({∅} ∪ {1𝑜})
64, 5eqtr4i 2108 . 2 (1𝑜 ∪ {1𝑜}) = {∅, 1𝑜}
71, 2, 63eqtri 2109 1 2𝑜 = {∅, 1𝑜}
Colors of variables: wff set class
Syntax hints:   = wceq 1287  cun 2986  c0 3275  {csn 3431  {cpr 3432  suc csuc 4168  1𝑜c1o 6130  2𝑜c2o 6131
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067
This theorem depends on definitions:  df-bi 115  df-tru 1290  df-nf 1393  df-sb 1690  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-dif 2990  df-un 2992  df-nul 3276  df-pr 3438  df-suc 4174  df-1o 6137  df-2o 6138
This theorem is referenced by:  df2o2  6152  2oconcl  6159  en2eqpr  6577  finomni  6743  exmidomniim  6744  exmidomni  6745  fodjuomnilemf  6747  exmidfodomrlemr  6775  exmidfodomrlemrALT  6776  0lt2o  11354  1lt2o  11355  el2oss1o  11356  nninfalllem1  11368  nninfall  11369  nninfsellemqall  11376  nninfomnilem  11379
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