Theorem df3o2 43275

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 8524	. 2 ⊢ 3o = suc 2o
2		df2o3 8530	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4187	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6401	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4653	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2778	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2768	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1537   ∪ cun 3974  ∅c0 4352  {csn 4648  {cpr 4650  {ctp 4652  suc csuc 6397  1_oc1o 8515  2_oc2o 8516  3_oc3o 8517

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-v 3490  df-dif 3979  df-un 3981  df-nul 4353  df-pr 4651  df-tp 4653  df-suc 6401  df-1o 8522  df-2o 8523  df-3o 8524

This theorem is referenced by:  oenord1ex  43277  oenord1  43278  clsk1indlem4  44006  clsk1indlem1  44007