Theorem df2o2 7732

Description: Expanded value of the ordinal number 2. (Contributed by NM, 29-Jan-2004.)

Assertion

Ref	Expression
df2o2	⊢ 2𝑜 = {∅, {∅}}

Proof of Theorem df2o2

Step	Hyp	Ref	Expression
1		df2o3 7731	. 2 ⊢ 2𝑜 = {∅, 1𝑜}
2		df1o2 7730	. . 3 ⊢ 1𝑜 = {∅}
3	2	preq2i 4409	. 2 ⊢ {∅, 1𝑜} = {∅, {∅}}
4	1, 3	eqtri 2793	1 ⊢ 2𝑜 = {∅, {∅}}

Syntax hints:   = wceq 1631  ∅c0 4063  {csn 4317  {cpr 4319  1_𝑜c1o 7710  2_𝑜c2o 7711

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-sn 4318  df-pr 4320  df-suc 5871  df-1o 7717  df-2o 7718

This theorem is referenced by:  2dom  8186  pw2eng  8226  pwcda1  9222  canthp1lem1  9680  pr0hash2ex  13398  hashpw  13425  znidomb  20125  ssoninhaus  32784  onint1  32785  pw2f1ocnv  38130  df3o3  38849