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Theorem orddif 4678
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 4622 . 2  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
2 disj3 3674 . . 3  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( A  \  { A } ) )
3 df-suc 4590 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
43difeq1i 3463 . . . . 5  |-  ( suc 
A  \  { A } )  =  ( ( A  u.  { A } )  \  { A } )
5 difun2 3709 . . . . 5  |-  ( ( A  u.  { A } )  \  { A } )  =  ( A  \  { A } )
64, 5eqtri 2458 . . . 4  |-  ( suc 
A  \  { A } )  =  ( A  \  { A } )
76eqeq2i 2448 . . 3  |-  ( A  =  ( suc  A  \  { A } )  <-> 
A  =  ( A 
\  { A }
) )
82, 7bitr4i 245 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( suc  A  \  { A } ) )
91, 8sylib 190 1  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1653    \ cdif 3319    u. cun 3320    i^i cin 3321   (/)c0 3630   {csn 3816   Ord word 4583   suc csuc 4586
This theorem is referenced by:  phplem3  7291  phplem4  7292  pssnn  7330
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-sbc 3164  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-br 4216  df-opab 4270  df-eprel 4497  df-fr 4544  df-we 4546  df-ord 4587  df-suc 4590
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