Theorem df3o3 43310

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 8439	. 2 ⊢ 3o = suc 2o
2		df2o2 8446	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4603	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4132	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6341	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4597	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2763	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2753	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1540   ∪ cun 3915  ∅c0 4299  {csn 4592  {cpr 4594  {ctp 4596  suc csuc 6337  2_oc2o 8431  3_oc3o 8432

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3920  df-un 3922  df-nul 4300  df-sn 4593  df-pr 4595  df-tp 4597  df-suc 6341  df-1o 8437  df-2o 8438  df-3o 8439

This theorem is referenced by: (None)