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Theorem orddif 5533
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 5478 . 2  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
2 disj3 3838 . . 3  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( A  \  { A } ) )
3 df-suc 5446 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
43difeq1i 3580 . . . . 5  |-  ( suc 
A  \  { A } )  =  ( ( A  u.  { A } )  \  { A } )
5 difun2 3876 . . . . 5  |-  ( ( A  u.  { A } )  \  { A } )  =  ( A  \  { A } )
64, 5eqtri 2452 . . . 4  |-  ( suc 
A  \  { A } )  =  ( A  \  { A } )
76eqeq2i 2441 . . 3  |-  ( A  =  ( suc  A  \  { A } )  <-> 
A  =  ( A 
\  { A }
) )
82, 7bitr4i 256 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( suc  A  \  { A } ) )
91, 8sylib 200 1  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1438    \ cdif 3434    u. cun 3435    i^i cin 3436   (/)c0 3762   {csn 3997   Ord word 5439   suc csuc 5442
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-9 1873  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-sep 4544  ax-nul 4553  ax-pr 4658
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 985  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-eu 2270  df-mo 2271  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-ral 2781  df-rex 2782  df-rab 2785  df-v 3084  df-sbc 3301  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3763  df-if 3911  df-sn 3998  df-pr 4000  df-op 4004  df-br 4422  df-opab 4481  df-eprel 4762  df-fr 4810  df-we 4812  df-ord 5443  df-suc 5446
This theorem is referenced by:  phplem3  7757  phplem4  7758  pssnn  7794
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