Theorem df3o2 43670

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 8409	. 2 ⊢ 3o = suc 2o
2		df2o3 8415	. . . 4 ⊢ 2o = {∅, 1o}
3	2	uneq1i 4118	. . 3 ⊢ (2o ∪ {2o}) = ({∅, 1o} ∪ {2o})
4		df-suc 6331	. . 3 ⊢ suc 2o = (2o ∪ {2o})
5		df-tp 4587	. . 3 ⊢ {∅, 1o, 2o} = ({∅, 1o} ∪ {2o})
6	3, 4, 5	3eqtr4i 2770	. 2 ⊢ suc 2o = {∅, 1o, 2o}
7	1, 6	eqtri 2760	1 ⊢ 3o = {∅, 1o, 2o}

Syntax hints:   = wceq 1542   ∪ cun 3901  ∅c0 4287  {csn 4582  {cpr 4584  {ctp 4586  suc csuc 6327  1_oc1o 8400  2_oc2o 8401  3_oc3o 8402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-dif 3906  df-un 3908  df-nul 4288  df-pr 4585  df-tp 4587  df-suc 6331  df-1o 8407  df-2o 8408  df-3o 8409

This theorem is referenced by:  oenord1ex  43672  oenord1  43673  clsk1indlem4  44400  clsk1indlem1  44401