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Theorem ordnbtwn 4954
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 4884 . . 3  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
2 ordirr 4882 . . 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 4930 . . . . 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 865 . . . 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 2514 . . . . 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 1381    e. wcel 1802   Ord word 4863   suc csuc 4866
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-sbc 3312  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-tr 4527  df-eprel 4777  df-fr 4824  df-we 4826  df-ord 4867  df-suc 4870
This theorem is referenced by:  onnbtwn  4955  ordsucss  6634
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