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Theorem ordn2lp 4296
Description: An ordinal class cannot be an element of one of its members. Variant of first part of Theorem 2.2(vii) of [BellMachover] p. 469. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
ordn2lp  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)

Proof of Theorem ordn2lp
StepHypRef Expression
1 ordirr 4293 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 ordtr 4141 . . 3  |-  ( Ord 
A  ->  Tr  A
)
3 trel 3890 . . 3  |-  ( Tr  A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
42, 3syl 14 . 2  |-  ( Ord 
A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
51, 4mtod 622 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    e. wcel 1434   Tr wtr 3883   Ord word 4125
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4288
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-sn 3412  df-uni 3610  df-tr 3884  df-iord 4129
This theorem is referenced by: (None)
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