Theorem df3o3 43933

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 8455	. 2 ⊢ 3o = suc 2o
2		df2o2 8462	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4605	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4128	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6367	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4599	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2802	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2792	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1567   ∪ cun 3911  ∅c0 4294  {csn 4594  {cpr 4596  {ctp 4598  suc csuc 6363  2_oc2o 8447  3_oc3o 8448

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-nul 4295  df-sn 4595  df-pr 4597  df-tp 4599  df-suc 6367  df-1o 8453  df-2o 8454  df-3o 8455

This theorem is referenced by: (None)