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Theorem ordn2lp 5462
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 5460 . 2  |-  ( Ord 
A  ->  -.  A  e.  A )
2 ordtr 5456 . . 3  |-  ( Ord 
A  ->  Tr  A
)
3 trel 4527 . . 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 180 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 370    e. wcel 1870   Tr wtr 4520   Ord word 5441
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1751  ax-6 1797  ax-7 1841  ax-9 1874  ax-10 1889  ax-11 1894  ax-12 1907  ax-13 2055  ax-ext 2407  ax-sep 4548  ax-nul 4556  ax-pr 4661
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1790  df-eu 2270  df-mo 2271  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2579  df-ne 2627  df-ral 2787  df-rex 2788  df-rab 2791  df-v 3089  df-sbc 3306  df-dif 3445  df-un 3447  df-in 3449  df-ss 3456  df-nul 3768  df-if 3916  df-sn 4003  df-pr 4005  df-op 4009  df-uni 4223  df-br 4427  df-opab 4485  df-tr 4521  df-eprel 4765  df-fr 4813  df-we 4815  df-ord 5445
This theorem is referenced by:  ordtri1  5475  ordnbtwn  5532  suc11  5545  smoord  7092  unblem1  7829  cantnfp1lem3  8184  cardprclem  8412
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