Theorem df3o3 43671

Description: Ordinal 3, fully expanded. (Contributed by RP, 8-Jul-2021.)

Assertion

Ref	Expression
df3o3	⊢ 3o = {∅, {∅}, {∅, {∅}}}

Proof of Theorem df3o3

Step	Hyp	Ref	Expression
1		df-3o 8409	. 2 ⊢ 3o = suc 2o
2		df2o2 8416	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4593	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4120	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6331	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4587	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2770	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2760	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1542   ∪ cun 3901  ∅c0 4287  {csn 4582  {cpr 4584  {ctp 4586  suc csuc 6327  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-sn 4583  df-pr 4585  df-tp 4587  df-suc 6331  df-1o 8407  df-2o 8408  df-3o 8409

This theorem is referenced by: (None)