Theorem df3o2 44154

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 8457	. 2 ⊢ 3o = suc 2o
2		df2o3 8463	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4111	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6363	. . 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 1570   ∪ cun 3897  ∅c0 4279  {csn 4584  {cpr 4586  {ctp 4588  suc csuc 6359  1_oc1o 8448  2_oc2o 8449  3_oc3o 8450

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

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 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-v 3452  df-dif 3902  df-un 3904  df-nul 4280  df-pr 4587  df-tp 4589  df-suc 6363  df-1o 8455  df-2o 8456  df-3o 8457

This theorem is used by:  oenord1ex  44156  oenord1  44157  clsk1indlem4  44884  clsk1indlem1  44885