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Theorem orddif 4910
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 4855 . 2  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
2 disj3 3821 . . 3  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( A  \  { A } ) )
3 df-suc 4823 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
43difeq1i 3568 . . . . 5  |-  ( suc 
A  \  { A } )  =  ( ( A  u.  { A } )  \  { A } )
5 difun2 3856 . . . . 5  |-  ( ( A  u.  { A } )  \  { A } )  =  ( A  \  { A } )
64, 5eqtri 2480 . . . 4  |-  ( suc 
A  \  { A } )  =  ( A  \  { A } )
76eqeq2i 2469 . . 3  |-  ( A  =  ( suc  A  \  { A } )  <-> 
A  =  ( A 
\  { A }
) )
82, 7bitr4i 252 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( suc  A  \  { A } ) )
91, 8sylib 196 1  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1370    \ cdif 3423    u. cun 3424    i^i cin 3425   (/)c0 3735   {csn 3975   Ord word 4816   suc csuc 4819
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4511  ax-nul 4519  ax-pr 4629
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3070  df-sbc 3285  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-nul 3736  df-if 3890  df-sn 3976  df-pr 3978  df-op 3982  df-br 4391  df-opab 4449  df-eprel 4730  df-fr 4777  df-we 4779  df-ord 4820  df-suc 4823
This theorem is referenced by:  phplem3  7592  phplem4  7593  pssnn  7632
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