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Theorem ordnbtwn 4963
Description: There is no set between an ordinal class and its successor. Generalized Proposition 7.25 of [TakeutiZaring] p. 41. (Contributed by NM, 21-Jun-1998.)
Assertion
Ref Expression
ordnbtwn  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )

Proof of Theorem ordnbtwn
StepHypRef Expression
1 ordn2lp 4893 . . 3  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
2 ordirr 4891 . . 3  |-  ( Ord 
A  ->  -.  A  e.  A )
3 ioran 490 . . 3  |-  ( -.  ( ( A  e.  B  /\  B  e.  A )  \/  A  e.  A )  <->  ( -.  ( A  e.  B  /\  B  e.  A
)  /\  -.  A  e.  A ) )
41, 2, 3sylanbrc 664 . 2  |-  ( Ord 
A  ->  -.  (
( A  e.  B  /\  B  e.  A
)  \/  A  e.  A ) )
5 elsuci 4939 . . . . 5  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
65anim2i 569 . . . 4  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( A  e.  B  /\  ( B  e.  A  \/  B  =  A ) ) )
7 andi 863 . . . 4  |-  ( ( A  e.  B  /\  ( B  e.  A  \/  B  =  A
) )  <->  ( ( A  e.  B  /\  B  e.  A )  \/  ( A  e.  B  /\  B  =  A
) ) )
86, 7sylib 196 . . 3  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( ( A  e.  B  /\  B  e.  A )  \/  ( A  e.  B  /\  B  =  A )
) )
9 eleq2 2535 . . . . 5  |-  ( B  =  A  ->  ( A  e.  B  <->  A  e.  A ) )
109biimpac 486 . . . 4  |-  ( ( A  e.  B  /\  B  =  A )  ->  A  e.  A )
1110orim2i 518 . . 3  |-  ( ( ( A  e.  B  /\  B  e.  A
)  \/  ( A  e.  B  /\  B  =  A ) )  -> 
( ( A  e.  B  /\  B  e.  A )  \/  A  e.  A ) )
128, 11syl 16 . 2  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( ( A  e.  B  /\  B  e.  A )  \/  A  e.  A ) )
134, 12nsyl 121 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 368    /\ wa 369    = wceq 1374    e. wcel 1762   Ord word 4872   suc csuc 4875
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1963  ax-ext 2440  ax-sep 4563  ax-nul 4571  ax-pr 4681
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2274  df-mo 2275  df-clab 2448  df-cleq 2454  df-clel 2457  df-nfc 2612  df-ne 2659  df-ral 2814  df-rex 2815  df-rab 2818  df-v 3110  df-sbc 3327  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3781  df-if 3935  df-sn 4023  df-pr 4025  df-op 4029  df-uni 4241  df-br 4443  df-opab 4501  df-tr 4536  df-eprel 4786  df-fr 4833  df-we 4835  df-ord 4876  df-suc 4879
This theorem is referenced by:  onnbtwn  4964  ordsucss  6626
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