Theorem df3o2 43269

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 8477	. 2 ⊢ 3o = suc 2o
2		df2o3 8483	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4137	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6356	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4604	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2767	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2757	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1539   ∪ cun 3922  ∅c0 4306  {csn 4599  {cpr 4601  {ctp 4603  suc csuc 6352  1_oc1o 8468  2_oc2o 8469  3_oc3o 8470

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-v 3459  df-dif 3927  df-un 3929  df-nul 4307  df-pr 4602  df-tp 4604  df-suc 6356  df-1o 8475  df-2o 8476  df-3o 8477

This theorem is referenced by:  oenord1ex  43271  oenord1  43272  clsk1indlem4  44000  clsk1indlem1  44001