Theorem df3o3 43891

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 4593	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4119	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6352	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4587	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2795	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2785	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1560   ∪ cun 3902  ∅c0 4285  {csn 4582  {cpr 4584  {ctp 4586  suc csuc 6348  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 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4583  df-pr 4585  df-tp 4587  df-suc 6352  df-1o 8437  df-2o 8438  df-3o 8439

This theorem is referenced by: (None)