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Theorem orddif 4667
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
StepHypRef Expression
1 orddisj 4611 . 2  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
2 disj3 3664 . . 3  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( A  \  { A } ) )
3 df-suc 4579 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
43difeq1i 3453 . . . . 5  |-  ( suc 
A  \  { A } )  =  ( ( A  u.  { A } )  \  { A } )
5 difun2 3699 . . . . 5  |-  ( ( A  u.  { A } )  \  { A } )  =  ( A  \  { A } )
64, 5eqtri 2455 . . . 4  |-  ( suc 
A  \  { A } )  =  ( A  \  { A } )
76eqeq2i 2445 . . 3  |-  ( A  =  ( suc  A  \  { A } )  <-> 
A  =  ( A 
\  { A }
) )
82, 7bitr4i 244 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( suc  A  \  { A } ) )
91, 8sylib 189 1  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1652    \ cdif 3309    u. cun 3310    i^i cin 3311   (/)c0 3620   {csn 3806   Ord word 4572   suc csuc 4575
This theorem is referenced by:  phplem3  7280  phplem4  7281  pssnn  7319
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-eprel 4486  df-fr 4533  df-we 4535  df-ord 4576  df-suc 4579
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