Theorem df3o3 37343

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 7449	. 2 ⊢ 3𝑜 = suc 2𝑜
2		df2o2 7461	. . . 4 ⊢ 2𝑜 = {∅, {∅}}
3	2	sneqi 4136	. . . 4 ⊢ {2𝑜} = {{∅, {∅}}}
4	2, 3	uneq12i 3727	. . 3 ⊢ (2𝑜 ∪ {2𝑜}) = ({∅, {∅}} ∪ {{∅, {∅}}})
5		df-suc 5646	. . 3 ⊢ suc 2𝑜 = (2𝑜 ∪ {2𝑜})
6		df-tp 4130	. . 3 ⊢ {∅, {∅}, {∅, {∅}}} = ({∅, {∅}} ∪ {{∅, {∅}}})
7	4, 5, 6	3eqtr4i 2642	. 2 ⊢ suc 2𝑜 = {∅, {∅}, {∅, {∅}}}
8	1, 7	eqtri 2632	1 ⊢ 3𝑜 = {∅, {∅}, {∅, {∅}}}

Syntax hints:   = wceq 1475   ∪ cun 3538  ∅c0 3874  {csn 4125  {cpr 4127  {ctp 4129  suc csuc 5642  2_𝑜c2o 7441  3_𝑜c3o 7442

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-un 3545  df-nul 3875  df-sn 4126  df-pr 4128  df-tp 4130  df-suc 5646  df-1o 7447  df-2o 7448  df-3o 7449

This theorem is referenced by: (None)