Theorem df3o3 41635

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 8299	. 2 ⊢ 3o = suc 2o
2		df2o2 8306	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4572	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4095	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6272	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4566	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2776	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2766	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1539   ∪ cun 3885  ∅c0 4256  {csn 4561  {cpr 4563  {ctp 4565  suc csuc 6268  2_oc2o 8291  3_oc3o 8292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-un 3892  df-nul 4257  df-sn 4562  df-pr 4564  df-tp 4566  df-suc 6272  df-1o 8297  df-2o 8298  df-3o 8299

This theorem is referenced by: (None)