MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df2o2 Structured version   Visualization version   GIF version

Theorem df2o2 7571
Description: Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)
Assertion
Ref Expression
df2o2 2𝑜 = {∅, {∅}}

Proof of Theorem df2o2
StepHypRef Expression
1 df2o3 7570 . 2 2𝑜 = {∅, 1𝑜}
2 df1o2 7569 . . 3 1𝑜 = {∅}
32preq2i 4270 . 2 {∅, 1𝑜} = {∅, {∅}}
41, 3eqtri 2643 1 2𝑜 = {∅, {∅}}
Colors of variables: wff setvar class
Syntax hints:   = wceq 1482  c0 3913  {csn 4175  {cpr 4177  1𝑜c1o 7550  2𝑜c2o 7551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-v 3200  df-dif 3575  df-un 3577  df-nul 3914  df-sn 4176  df-pr 4178  df-suc 5727  df-1o 7557  df-2o 7558
This theorem is referenced by:  2dom  8026  pw2eng  8063  pwcda1  9013  canthp1lem1  9471  pr0hash2ex  13191  hashpw  13218  znidomb  19904  ssoninhaus  32431  onint1  32432  pw2f1ocnv  37430  df3o3  38149
  Copyright terms: Public domain W3C validator