Theorem df3o3 42527

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 8474	. 2 ⊢ 3o = suc 2o
2		df2o2 8481	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4639	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4161	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6370	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4633	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2769	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2759	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1540   ∪ cun 3946  ∅c0 4322  {csn 4628  {cpr 4630  {ctp 4632  suc csuc 6366  2_oc2o 8466  3_oc3o 8467

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2702

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-fal 1553  df-ex 1781  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-v 3475  df-dif 3951  df-un 3953  df-nul 4323  df-sn 4629  df-pr 4631  df-tp 4633  df-suc 6370  df-1o 8472  df-2o 8473  df-3o 8474

This theorem is referenced by: (None)