Theorem df3o2 41093

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 8115	. 2 ⊢ 3o = suc 2o
2		df2o3 8128	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4065	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6176	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4528	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2792	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2782	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1539   ∪ cun 3857  ∅c0 4226  {csn 4523  {cpr 4525  {ctp 4527  suc csuc 6172  1_oc1o 8106  2_oc2o 8107  3_oc3o 8108

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2730

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-tru 1542  df-ex 1783  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-v 3412  df-dif 3862  df-un 3864  df-nul 4227  df-pr 4526  df-tp 4528  df-suc 6176  df-1o 8113  df-2o 8114  df-3o 8115

This theorem is referenced by:  clsk1indlem4  41113  clsk1indlem1  41114