Theorem df3o3 43498

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 8397	. 2 ⊢ 3o = suc 2o
2		df2o2 8404	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4589	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4116	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6321	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4583	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2767	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2757	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1541   ∪ cun 3897  ∅c0 4283  {csn 4578  {cpr 4580  {ctp 4582  suc csuc 6317  2_oc2o 8389  3_oc3o 8390

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 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-v 3440  df-dif 3902  df-un 3904  df-nul 4284  df-sn 4579  df-pr 4581  df-tp 4583  df-suc 6321  df-1o 8395  df-2o 8396  df-3o 8397

This theorem is referenced by: (None)