Theorem df2o2 7812

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 7811	. 2 ⊢ 2𝑜 = {∅, 1𝑜}
2		df1o2 7810	. . 3 ⊢ 1𝑜 = {∅}
3	2	preq2i 4459	. 2 ⊢ {∅, 1𝑜} = {∅, {∅}}
4	1, 3	eqtri 2819	1 ⊢ 2𝑜 = {∅, {∅}}

Syntax hints:   = wceq 1653  ∅c0 4113  {csn 4366  {cpr 4368  1_𝑜c1o 7790  2_𝑜c2o 7791

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-ext 2775

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2784  df-cleq 2790  df-clel 2793  df-nfc 2928  df-v 3385  df-dif 3770  df-un 3772  df-nul 4114  df-sn 4367  df-pr 4369  df-suc 5945  df-1o 7797  df-2o 7798

This theorem is referenced by:  2dom  8266  pw2eng  8306  pwcda1  9302  canthp1lem1  9760  pr0hash2ex  13441  hashpw  13468  znidomb  20228  ssoninhaus  32947  onint1  32948  pw2f1ocnv  38377  df3o3  39093