Theorem orddif 4524

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 4523	. 2 |- ( Ord A -> ( A i^i { A } ) = (/) )
2		disj3 3461	. . 3 |- ( ( A i^i { A } ) = (/) <-> A = ( A \ { A } ) )
3		df-suc 4349	. . . . . 6 |- suc A = ( A u. { A } )
4	3	difeq1i 3236	. . . . 5 |- ( suc A \ { A } ) = ( ( A u. { A } ) \ { A } )
5		difun2 3488	. . . . 5 |- ( ( A u. { A } ) \ { A } ) = ( A \ { A } )
6	4, 5	eqtri 2186	. . . 4 |- ( suc A \ { A } ) = ( A \ { A } )
7	6	eqeq2i 2176	. . 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 1343   \ cdif 3113   u. cun 3114   i^i cin 3115   (/)c0 3409   {csn 3576   Ord word 4340   suc csuc 4343

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-setind 4514

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-sn 3582  df-suc 4349

This theorem is referenced by:  phplem3  6820  phplem4  6821  phplem4dom  6828  phplem4on  6833  dif1en  6845