Theorem df3o3 43774

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 8401	. 2 ⊢ 3o = suc 2o
2		df2o2 8408	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4569	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4099	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6320	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4563	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2774	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2764	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1548   ∪ cun 3883  ∅c0 4264  {csn 4558  {cpr 4560  {ctp 4562  suc csuc 6316  2_oc2o 8393  3_oc3o 8394

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-v 3435  df-dif 3888  df-un 3890  df-nul 4265  df-sn 4559  df-pr 4561  df-tp 4563  df-suc 6320  df-1o 8399  df-2o 8400  df-3o 8401

This theorem is referenced by: (None)