Theorem df3o3 42366

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 8470	. 2 ⊢ 3o = suc 2o
2		df2o2 8477	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4638	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4160	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6369	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4632	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2768	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2758	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1539   ∪ cun 3945  ∅c0 4321  {csn 4627  {cpr 4629  {ctp 4631  suc csuc 6365  2_oc2o 8462  3_oc3o 8463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-un 3952  df-nul 4322  df-sn 4628  df-pr 4630  df-tp 4632  df-suc 6369  df-1o 8468  df-2o 8469  df-3o 8470

This theorem is referenced by: (None)