Theorem df3o2 43851

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 8433	. 2 ⊢ 3o = suc 2o
2		df2o3 8439	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4115	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6347	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4584	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2794	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2784	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1559   ∪ cun 3900  ∅c0 4283  {csn 4579  {cpr 4581  {ctp 4583  suc csuc 6343  1_oc1o 8424  2_oc2o 8425  3_oc3o 8426

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-dif 3905  df-un 3907  df-nul 4284  df-pr 4582  df-tp 4584  df-suc 6347  df-1o 8431  df-2o 8432  df-3o 8433

This theorem is referenced by:  oenord1ex  43853  oenord1  43854  clsk1indlem4  44581  clsk1indlem1  44582