Theorem df3o2 42365

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 8470	. 2 ⊢ 3o = suc 2o
2		df2o3 8476	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4158	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6369	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4632	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2768	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2758	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1539   ∪ cun 3945  ∅c0 4321  {csn 4627  {cpr 4629  {ctp 4631  suc csuc 6365  1_oc1o 8461  2_oc2o 8462  3_oc3o 8463

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-v 3474  df-dif 3950  df-un 3952  df-nul 4322  df-pr 4630  df-tp 4632  df-suc 6369  df-1o 8468  df-2o 8469  df-3o 8470

This theorem is referenced by:  oenord1ex  42367  oenord1  42368  clsk1indlem4  43097  clsk1indlem1  43098