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Theorem ordn2lp 5442
Description: An ordinal class cannot 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 5440 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 ordtr 5436 . . 3  |-  ( Ord 
A  ->  Tr  A
)
3 trel 4503 . . 3  |-  ( Tr  A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
42, 3syl 17 . 2  |-  ( Ord 
A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
51, 4mtod 181 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 371    e. wcel 1886   Tr wtr 4496   Ord word 5421
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pr 4638
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-nul 3731  df-if 3881  df-sn 3968  df-pr 3970  df-op 3974  df-uni 4198  df-br 4402  df-opab 4461  df-tr 4497  df-eprel 4744  df-fr 4792  df-we 4794  df-ord 5425
This theorem is referenced by:  ordtri1  5455  ordnbtwn  5512  suc11  5525  smoord  7081  unblem1  7820  cantnfp1lem3  8182  cardprclem  8410
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