Theorem df3o2 42552

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 8463	. 2 ⊢ 3o = suc 2o
2		df2o3 8469	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4151	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6360	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4625	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2762	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2752	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1533   ∪ cun 3938  ∅c0 4314  {csn 4620  {cpr 4622  {ctp 4624  suc csuc 6356  1_oc1o 8454  2_oc2o 8455  3_oc3o 8456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-dif 3943  df-un 3945  df-nul 4315  df-pr 4623  df-tp 4625  df-suc 6360  df-1o 8461  df-2o 8462  df-3o 8463

This theorem is referenced by:  oenord1ex  42554  oenord1  42555  clsk1indlem4  43284  clsk1indlem1  43285