Theorem df3o2 43551

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 8399	. 2 ⊢ 3o = suc 2o
2		df2o3 8405	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4116	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6323	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4585	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2769	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2759	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1541   ∪ cun 3899  ∅c0 4285  {csn 4580  {cpr 4582  {ctp 4584  suc csuc 6319  1_oc1o 8390  2_oc2o 8391  3_oc3o 8392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3442  df-dif 3904  df-un 3906  df-nul 4286  df-pr 4583  df-tp 4585  df-suc 6323  df-1o 8397  df-2o 8398  df-3o 8399

This theorem is referenced by:  oenord1ex  43553  oenord1  43554  clsk1indlem4  44281  clsk1indlem1  44282