Theorem df3o3 40368

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 8098	. 2 ⊢ 3o = suc 2o
2		df2o2 8112	. . . 4 ⊢ 2o = {∅, {∅}}
3	2	sneqi 4572	. . . 4 ⊢ {2o} = {{∅, {∅}}}
4	2, 3	uneq12i 4137	. . 3 ⊢ (2o ∪ {2o}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 6192	. . 3 ⊢ suc 2o = (2o ∪ {2o})
6		df-tp 4566	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2854	. 2 ⊢ suc 2o = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2844	1 ⊢ 3o = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1533   ∪ cun 3934  ∅c0 4291  {csn 4561  {cpr 4563  {ctp 4565  suc csuc 6188  2_oc2o 8090  3_oc3o 8091

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-v 3497  df-dif 3939  df-un 3941  df-nul 4292  df-sn 4562  df-pr 4564  df-tp 4566  df-suc 6192  df-1o 8096  df-2o 8097  df-3o 8098

This theorem is referenced by: (None)