Theorem orddif 5820

Description: Ordinal derived from its successor. (Contributed by NM, 20-May-1998.)

Assertion

Ref	Expression
orddif	|- ( Ord A -> A = ( suc A \ { A } ) )

Proof of Theorem orddif

Step	Hyp	Ref	Expression
1		orddisj 5762	. 2 |- ( Ord A -> ( A i^i { A } ) = (/) )
2		disj3 4021	. . 3 |- ( ( A i^i { A } ) = (/) <-> A = ( A \ { A } ) )
3		df-suc 5729	. . . . . 6 |- suc A = ( A u. { A } )
4	3	difeq1i 3724	. . . . 5 |- ( suc A \ { A } ) = ( ( A u. { A } ) \ { A } )
5		difun2 4048	. . . . 5 |- ( ( A u. { A } ) \ { A } ) = ( A \ { A } )
6	4, 5	eqtri 2644	. . . 4 |- ( suc A \ { A } ) = ( A \ { A } )
7	6	eqeq2i 2634	. . 3 |- ( A = ( suc A \ { A } ) <-> A = ( A \ { A } ) )
8	2, 7	bitr4i 267	. 2 |- ( ( A i^i { A } ) = (/) <-> A = ( suc A \ { A } ) )
9	1, 8	sylib 208	1 |- ( Ord A -> A = ( suc A \ { A } ) )

Syntax hints:   -> wi 4   = wceq 1483   \ cdif 3571   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   Ord word 5722   suc csuc 5725

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  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-eprel 5029  df-fr 5073  df-we 5075  df-ord 5726  df-suc 5729

This theorem is referenced by:  phplem3  8141  phplem4  8142  pssnn  8178