Theorem df3o2 43741

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 8407	. 2 ⊢ 3o = suc 2o
2		df2o3 8413	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4104	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6329	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4572	. . 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 1542   ∪ cun 3887  ∅c0 4273  {csn 4567  {cpr 4569  {ctp 4571  suc csuc 6325  1_oc1o 8398  2_oc2o 8399  3_oc3o 8400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-dif 3892  df-un 3894  df-nul 4274  df-pr 4570  df-tp 4572  df-suc 6329  df-1o 8405  df-2o 8406  df-3o 8407

This theorem is referenced by:  oenord1ex  43743  oenord1  43744  clsk1indlem4  44471  clsk1indlem1  44472