Theorem df3o3 44061

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 8451	. 2 ⊢ 3o = suc 2o
2		df2o2 8458	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4600	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4120	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6366	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4594	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2796	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2786	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1570   ∪ cun 3903  ∅c0 4286  {csn 4589  {cpr 4591  {ctp 4593  suc csuc 6362  2_oc2o 8443  3_oc3o 8444

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-v 3457  df-dif 3908  df-un 3910  df-nul 4287  df-sn 4590  df-pr 4592  df-tp 4594  df-suc 6366  df-1o 8449  df-2o 8450  df-3o 8451

This theorem is referenced by: (None)