Theorem orddif 4583

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 4582	. 2 |- ( Ord A -> ( A i^i { A } ) = (/) )
2		disj3 3503	. . 3 |- ( ( A i^i { A } ) = (/) <-> A = ( A \ { A } ) )
3		df-suc 4406	. . . . . 6 |- suc A = ( A u. { A } )
4	3	difeq1i 3277	. . . . 5 |- ( suc A \ { A } ) = ( ( A u. { A } ) \ { A } )
5		difun2 3530	. . . . 5 |- ( ( A u. { A } ) \ { A } ) = ( A \ { A } )
6	4, 5	eqtri 2217	. . . 4 |- ( suc A \ { A } ) = ( A \ { A } )
7	6	eqeq2i 2207	. . 3 |- ( A = ( suc A \ { A } ) <-> A = ( A \ { A } ) )
8	2, 7	bitr4i 187	. 2 |- ( ( A i^i { A } ) = (/) <-> A = ( suc A \ { A } ) )
9	1, 8	sylib 122	1 |- ( Ord A -> A = ( suc A \ { A } ) )

Syntax hints:   -> wi 4   = wceq 1364   \ cdif 3154   u. cun 3155   i^i cin 3156   (/)c0 3450   {csn 3622   Ord word 4397   suc csuc 4400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178  ax-setind 4573

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-suc 4406

This theorem is referenced by:  phplem3  6915  phplem4  6916  phplem4dom  6923  phplem4on  6928  dif1en  6940