Theorem df3o3 38323

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 7562	. 2 |- 3o = suc 2o
2		df2o2 7574	. . . 4 |- 2o = { (/) , { (/) } }
3	2	sneqi 4188	. . . 4 |- { 2o } = { { (/) , { (/) } } }
4	2, 3	uneq12i 3765	. . 3 |- ( 2o u. { 2o } ) = ( { (/) , { (/) } } u. { { (/) , { (/) } } } )
5		df-suc 5729	. . 3 |- suc 2o = ( 2o u. { 2o } )
6		df-tp 4182	. . 3 |- { (/) , { (/) } , { (/) , { (/) } } } = ( { (/) , { (/) } } u. { { (/) , { (/) } } } )
7	4, 5, 6	3eqtr4i 2654	. 2 |- suc 2o = { (/) , { (/) } , { (/) , { (/) } } }
8	1, 7	eqtri 2644	1 |- 3o = { (/) , { (/) } , { (/) , { (/) } } }

Syntax hints:   = wceq 1483   u. cun 3572   (/)c0 3915   {csn 4177   {cpr 4179   {ctp 4181   suc csuc 5725   2oc2o 7554   3oc3o 7555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-un 3579  df-nul 3916  df-sn 4178  df-pr 4180  df-tp 4182  df-suc 5729  df-1o 7560  df-2o 7561  df-3o 7562

This theorem is referenced by: (None)