Theorem df3o3 41524

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 8269	. 2 ⊢ 3o = suc 2o
2		df2o2 8283	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4569	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4091	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6257	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4563	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2776	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2766	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1539   ∪ cun 3881  ∅c0 4253  {csn 4558  {cpr 4560  {ctp 4562  suc csuc 6253  2_oc2o 8261  3_oc3o 8262

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-sn 4559  df-pr 4561  df-tp 4563  df-suc 6257  df-1o 8267  df-2o 8268  df-3o 8269

This theorem is referenced by: (None)