Theorem df2o2 3112

Description: Expanded value of the ordinal number 2.

Assertion

Ref	Expression
df2o2	⊢ 2o = {∅, {∅}}

Proof of Theorem df2o2

Step	Hyp	Ref	Expression
1		df-2o 3105	. 2 ⊢ 2o = suc 1o
2		df-suc 2205	. 2 ⊢ suc 1o = (1o ∪ {1o})
3		df1o2 3111	. . . 4 ⊢ 1o = {∅}
4	3	sneqi 1817	. . . 4 ⊢ {1o} = {{∅}}
5	3, 4	uneq12i 1609	. . 3 ⊢ (1o ∪ {1o}) = ({∅} ∪ {{∅}})
6		df-pr 1812	. . 3 ⊢ {∅, {∅}} = ({∅} ∪ {{∅}})
7	5, 6	eqtr4 1122	. 2 ⊢ (1o ∪ {1o}) = {∅, {∅}}
8	1, 2, 7	3eqtr 1123	1 ⊢ 2o = {∅, {∅}}

Syntax hints:   = wceq 1091   ∪ cun 1485  ∅c0 1707  {csn 1808  {cpr 1809  suc csuc 2201  1_oc1o 3099  2_oc2o 3100

This theorem is referenced by:  2dom 3332  pw2en 3348  xp2cda 3723

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-nul 1708  df-sn 1811  df-pr 1812  df-suc 2205  df-1o 3104  df-2o 3105

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