Theorem df3o3 43276

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 8524	. 2 ⊢ 3o = suc 2o
2		df2o2 8531	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4659	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4189	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6401	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4653	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2778	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2768	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1537   ∪ cun 3974  ∅c0 4352  {csn 4648  {cpr 4650  {ctp 4652  suc csuc 6397  2_oc2o 8516  3_oc3o 8517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-sn 4649  df-pr 4651  df-tp 4653  df-suc 6401  df-1o 8522  df-2o 8523  df-3o 8524

This theorem is referenced by: (None)