Theorem orddif 4400

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 4399	. 2 |- ( Ord A -> ( A i^i { A } ) = (/) )
2		disj3 3362	. . 3 |- ( ( A i^i { A } ) = (/) <-> A = ( A \ { A } ) )
3		df-suc 4231	. . . . . 6 |- suc A = ( A u. { A } )
4	3	difeq1i 3137	. . . . 5 |- ( suc A \ { A } ) = ( ( A u. { A } ) \ { A } )
5		difun2 3389	. . . . 5 |- ( ( A u. { A } ) \ { A } ) = ( A \ { A } )
6	4, 5	eqtri 2120	. . . 4 |- ( suc A \ { A } ) = ( A \ { A } )
7	6	eqeq2i 2110	. . 3 |- ( A = ( suc A \ { A } ) <-> A = ( A \ { A } ) )
8	2, 7	bitr4i 186	. 2 |- ( ( A i^i { A } ) = (/) <-> A = ( suc A \ { A } ) )
9	1, 8	sylib 121	1 |- ( Ord A -> A = ( suc A \ { A } ) )

Syntax hints:   -> wi 4   = wceq 1299   \ cdif 3018   u. cun 3019   i^i cin 3020   (/)c0 3310   {csn 3474   Ord word 4222   suc csuc 4225

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-setind 4390

This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-ral 2380  df-rab 2384  df-v 2643  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-nul 3311  df-sn 3480  df-suc 4231

This theorem is referenced by:  phplem3  6677  phplem4  6678  phplem4dom  6685  phplem4on  6690  dif1en  6702