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Theorem orddisj 4560
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 4556 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 disjsn 3669 . 2  |-  ( ( A  i^i  { A } )  =  (/)  <->  -.  A  e.  A )
31, 2sylibr 134 1  |-  ( Ord 
A  ->  ( A  i^i  { A } )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1364    e. wcel 2160    i^i cin 3143   (/)c0 3437   {csn 3607   Ord word 4377
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4551
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-in 3150  df-nul 3438  df-sn 3613
This theorem is referenced by:  orddif  4561  phplem2  6871  ennnfonelemhf1o  12432
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