Theorem orddif 4531

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 4530	. 2 |- ( Ord A -> ( A i^i { A } ) = (/) )
2		disj3 3467	. . 3 |- ( ( A i^i { A } ) = (/) <-> A = ( A \ { A } ) )
3		df-suc 4356	. . . . . 6 |- suc A = ( A u. { A } )
4	3	difeq1i 3241	. . . . 5 |- ( suc A \ { A } ) = ( ( A u. { A } ) \ { A } )
5		difun2 3494	. . . . 5 |- ( ( A u. { A } ) \ { A } ) = ( A \ { A } )
6	4, 5	eqtri 2191	. . . 4 |- ( suc A \ { A } ) = ( A \ { A } )
7	6	eqeq2i 2181	. . 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 1348   \ cdif 3118   u. cun 3119   i^i cin 3120   (/)c0 3414   {csn 3583   Ord word 4347   suc csuc 4350

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rab 2457  df-v 2732  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-sn 3589  df-suc 4356

This theorem is referenced by:  phplem3  6832  phplem4  6833  phplem4dom  6840  phplem4on  6845  dif1en  6857