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Theorem orddisj 4905
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 4885 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 disjsn 4076 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  -.  A  e.  A )
31, 2sylibr 212 1  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1398    e. wcel 1823    i^i cin 3460   (/)c0 3783   {csn 4016   Ord word 4866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-br 4440  df-opab 4498  df-eprel 4780  df-fr 4827  df-we 4829  df-ord 4870
This theorem is referenced by:  orddif  4960  tfrlem10  7048  phplem2  7690  isinf  7726  pssnn  7731  dif1en  7745  ackbij1lem5  8595  ackbij1lem14  8604  ackbij1lem16  8606  unsnen  8919  pwfi2f1o  31283  pwfi2f1oOLD  31284
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