Theorem df3o3 43759

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 8397	. 2 ⊢ 3o = suc 2o
2		df2o2 8404	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4566	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4096	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6316	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4560	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2772	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2762	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1547   ∪ cun 3881  ∅c0 4261  {csn 4555  {cpr 4557  {ctp 4559  suc csuc 6312  2_oc2o 8389  3_oc3o 8390

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2711

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2718  df-cleq 2731  df-clel 2814  df-v 3433  df-dif 3886  df-un 3888  df-nul 4262  df-sn 4556  df-pr 4558  df-tp 4560  df-suc 6316  df-1o 8395  df-2o 8396  df-3o 8397

This theorem is referenced by: (None)