Theorem df3o3 43417

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 8387	. 2 ⊢ 3o = suc 2o
2		df2o2 8394	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4584	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4113	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6312	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4578	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2764	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2754	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1541   ∪ cun 3895  ∅c0 4280  {csn 4573  {cpr 4575  {ctp 4577  suc csuc 6308  2_oc2o 8379  3_oc3o 8380

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-v 3438  df-dif 3900  df-un 3902  df-nul 4281  df-sn 4574  df-pr 4576  df-tp 4578  df-suc 6312  df-1o 8385  df-2o 8386  df-3o 8387

This theorem is referenced by: (None)