Theorem df3o2 44576

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 8516	. 2 ⊢ 3o = suc 2o
2		df2o3 8522	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4166	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6394	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4593	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2796	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2786	1 ⊢ 3o = {∅, 1o, 2o}

This proof depends on syntax axioms:   = wceq 1570   ∪ cun 4152  ∅c0 4281  {csn 4588  {cpr 4590  {ctp 4592  suc csuc 6390  1_oc1o 8507  2_oc2o 8508  3_oc3o 8509

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-dif 4050  df-un 4154  df-nul 4282  df-pr 4591  df-tp 4593  df-suc 6394  df-1o 8514  df-2o 8515  df-3o 8516

This theorem is used by:  oenord1ex  44578  oenord1  44579  clsk1indlem4  45305  clsk1indlem1  45306