Theorem df3o2 43295

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 8438	. 2 ⊢ 3o = suc 2o
2		df2o3 8444	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4129	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6340	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4596	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2763	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2753	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1540   ∪ cun 3914  ∅c0 4298  {csn 4591  {cpr 4593  {ctp 4595  suc csuc 6336  1_oc1o 8429  2_oc2o 8430  3_oc3o 8431

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-v 3452  df-dif 3919  df-un 3921  df-nul 4299  df-pr 4594  df-tp 4596  df-suc 6340  df-1o 8436  df-2o 8437  df-3o 8438

This theorem is referenced by:  oenord1ex  43297  oenord1  43298  clsk1indlem4  44026  clsk1indlem1  44027