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Theorem ordn2lp 4739
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 4737 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 ordtr 4733 . . 3  |-  ( Ord 
A  ->  Tr  A
)
3 trel 4392 . . 3  |-  ( Tr  A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
42, 3syl 16 . 2  |-  ( Ord 
A  ->  ( ( A  e.  B  /\  B  e.  A )  ->  A  e.  A ) )
51, 4mtod 177 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1756   Tr wtr 4385   Ord word 4718
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2423  ax-sep 4413  ax-nul 4421  ax-pr 4531
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2257  df-mo 2258  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2720  df-rex 2721  df-rab 2724  df-v 2974  df-sbc 3187  df-dif 3331  df-un 3333  df-in 3335  df-ss 3342  df-nul 3638  df-if 3792  df-sn 3878  df-pr 3880  df-op 3884  df-uni 4092  df-br 4293  df-opab 4351  df-tr 4386  df-eprel 4632  df-fr 4679  df-we 4681  df-ord 4722
This theorem is referenced by:  ordtri1  4752  ordnbtwn  4809  suc11  4822  smoord  6826  unblem1  7564  cantnfp1lem3  7888  cantnfp1lem3OLD  7914  cardprclem  8149
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