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Theorem orddisj 4446
Description: An ordinal class and its singleton are disjoint. (Contributed by NM, 19-May-1998.)
Assertion
Ref Expression
orddisj  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )

Proof of Theorem orddisj
StepHypRef Expression
1 ordirr 4426 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 disjsn 3706 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  -.  A  e.  A )
31, 2sylibr 203 1  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1632    e. wcel 1696    i^i cin 3164   (/)c0 3468   {csn 3653   Ord word 4407
This theorem is referenced by:  orddif  4502  tfrlem10  6419  phplem2  7057  isinf  7092  pssnn  7097  dif1enOLD  7106  dif1en  7107  ackbij1lem5  7866  ackbij1lem14  7875  ackbij1lem16  7877  unsnen  8191  pwfi2f1o  27363
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
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