Theorem df3o3 43304

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 8507	. 2 ⊢ 3o = suc 2o
2		df2o2 8514	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4642	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4176	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6392	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4636	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2773	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2763	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1537   ∪ cun 3961  ∅c0 4339  {csn 4631  {cpr 4633  {ctp 4635  suc csuc 6388  2_oc2o 8499  3_oc3o 8500

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-v 3480  df-dif 3966  df-un 3968  df-nul 4340  df-sn 4632  df-pr 4634  df-tp 4636  df-suc 6392  df-1o 8505  df-2o 8506  df-3o 8507

This theorem is referenced by: (None)