Theorem df3o3 40728

Description: Ordinal 3, fully expanded. (Contributed by RP, 8-Jul-2021.)

Assertion

Ref	Expression
df3o3	⊢ 3o = {∅, {∅}, {∅, {∅}}}

Proof of Theorem df3o3

Step	Hyp	Ref	Expression
1		df-3o 8087	. 2 ⊢ 3o = suc 2o
2		df2o2 8101	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4536	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4088	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6165	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4530	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2831	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2821	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1538   ∪ cun 3879  ∅c0 4243  {csn 4525  {cpr 4527  {ctp 4529  suc csuc 6161  2_oc2o 8079  3_oc3o 8080

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-dif 3884  df-un 3886  df-nul 4244  df-sn 4526  df-pr 4528  df-tp 4530  df-suc 6165  df-1o 8085  df-2o 8086  df-3o 8087

This theorem is referenced by: (None)