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Theorem orddif 4502
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 4446 . 2  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
2 disj3 3512 . . 3  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( A  \  { A } ) )
3 df-suc 4414 . . . . . 6  |-  suc  A  =  ( A  u.  { A } )
43difeq1i 3303 . . . . 5  |-  ( suc 
A  \  { A } )  =  ( ( A  u.  { A } )  \  { A } )
5 difun2 3546 . . . . 5  |-  ( ( A  u.  { A } )  \  { A } )  =  ( A  \  { A } )
64, 5eqtri 2316 . . . 4  |-  ( suc 
A  \  { A } )  =  ( A  \  { A } )
76eqeq2i 2306 . . 3  |-  ( A  =  ( suc  A  \  { A } )  <-> 
A  =  ( A 
\  { A }
) )
82, 7bitr4i 243 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  A  =  ( suc  A  \  { A } ) )
91, 8sylib 188 1  |-  ( Ord 
A  ->  A  =  ( suc  A  \  { A } ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1632    \ cdif 3162    u. cun 3163    i^i cin 3164   (/)c0 3468   {csn 3653   Ord word 4407   suc csuc 4410
This theorem is referenced by:  phplem3  7058  phplem4  7059  pssnn  7097
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-eprel 4321  df-fr 4368  df-we 4370  df-ord 4411  df-suc 4414
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