Theorem df3o2 43927

Description: Ordinal 3 is the unordered triple containing ordinals 0, 1, and 2. (Contributed by RP, 8-Jul-2021.)

Assertion

Ref	Expression
df3o2	⊢ 3o = {∅, 1o, 2o}

Proof of Theorem df3o2

Step	Hyp	Ref	Expression
1		df-3o 8451	. 2 ⊢ 3o = suc 2o
2		df2o3 8457	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4126	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6364	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4596	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2802	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2792	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1567   ∪ cun 3911  ∅c0 4294  {csn 4591  {cpr 4593  {ctp 4595  suc csuc 6360  1_oc1o 8442  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 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-v 3465  df-dif 3916  df-un 3918  df-nul 4295  df-pr 4594  df-tp 4596  df-suc 6364  df-1o 8449  df-2o 8450  df-3o 8451

This theorem is referenced by:  oenord1ex  43929  oenord1  43930  clsk1indlem4  44657  clsk1indlem1  44658