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Theorem orddisj 4620
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 4600 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 disjsn 3869 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  -.  A  e.  A )
31, 2sylibr 205 1  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1653    e. wcel 1726    i^i cin 3320   (/)c0 3629   {csn 3815   Ord word 4581
This theorem is referenced by:  orddif  4676  tfrlem10  6649  phplem2  7288  isinf  7323  pssnn  7328  dif1enOLD  7341  dif1en  7342  ackbij1lem5  8105  ackbij1lem14  8114  ackbij1lem16  8116  unsnen  8429  pwfi2f1o  27238
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 2418  ax-sep 4331  ax-nul 4339  ax-pr 4404
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 2286  df-mo 2287  df-clab 2424  df-cleq 2430  df-clel 2433  df-nfc 2562  df-ne 2602  df-ral 2711  df-rex 2712  df-rab 2715  df-v 2959  df-sbc 3163  df-dif 3324  df-un 3326  df-in 3328  df-ss 3335  df-nul 3630  df-if 3741  df-sn 3821  df-pr 3822  df-op 3824  df-br 4214  df-opab 4268  df-eprel 4495  df-fr 4542  df-we 4544  df-ord 4585
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