Theorem df2o3 6037

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

Step	Hyp	Ref	Expression
1		df-2o 6025	. 2 ⊢ 2𝑜 = suc 1𝑜
2		df-suc 4126	. 2 ⊢ suc 1𝑜 = (1𝑜 ∪ {1𝑜})
3		df1o2 6036	. . . 4 ⊢ 1𝑜 = {∅}
4	3	uneq1i 3122	. . 3 ⊢ (1𝑜 ∪ {1𝑜}) = ({∅} ∪ {1𝑜})
5		df-pr 3405	. . 3 ⊢ {∅, 1𝑜} = ({∅} ∪ {1𝑜})
6	4, 5	eqtr4i 2104	. 2 ⊢ (1𝑜 ∪ {1𝑜}) = {∅, 1𝑜}
7	1, 2, 6	3eqtri 2105	1 ⊢ 2𝑜 = {∅, 1𝑜}

Syntax hints:   = wceq 1284   ∪ cun 2971  ∅c0 3251  {csn 3398  {cpr 3399  suc csuc 4120  1_𝑜c1o 6017  2_𝑜c2o 6018

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 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-dif 2975  df-un 2977  df-nul 3252  df-pr 3405  df-suc 4126  df-1o 6024  df-2o 6025

This theorem is referenced by:  df2o2  6038  2oconcl  6045  en2eqpr  6380