Theorem df3o2 40394

Description: Ordinal 3 is the triplet 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 8104	. 2 ⊢ 3o = suc 2o
2		df2o3 8117	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4135	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6197	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4572	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2854	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2844	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1537   ∪ cun 3934  ∅c0 4291  {csn 4567  {cpr 4569  {ctp 4571  suc csuc 6193  1_oc1o 8095  2_oc2o 8096  3_oc3o 8097

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3496  df-dif 3939  df-un 3941  df-nul 4292  df-pr 4570  df-tp 4572  df-suc 6197  df-1o 8102  df-2o 8103  df-3o 8104

This theorem is referenced by:  clsk1indlem4  40414  clsk1indlem1  40415