Theorem df3o2 44409

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 8482	. 2 ⊢ 3o = suc 2o
2		df2o3 8488	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4163	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6368	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4589	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2793	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2783	1 ⊢ 3o = {∅, 1o, 2o}

This proof depends on syntax axioms:   = wceq 1569   ∪ cun 4149  ∅c0 4278  {csn 4584  {cpr 4586  {ctp 4588  suc csuc 6364  1_oc1o 8473  2_oc2o 8474  3_oc3o 8475

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 4047  df-un 4151  df-nul 4279  df-pr 4587  df-tp 4589  df-suc 6368  df-1o 8480  df-2o 8481  df-3o 8482

This theorem is used by:  oenord1ex  44411  oenord1  44412  clsk1indlem4  45138  clsk1indlem1  45139