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Theorem ordnbtwn 4455
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 4384 . . 3  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
2 ordirr 4382 . . 3  |-  ( Ord 
A  ->  -.  A  e.  A )
3 ioran 478 . . 3  |-  ( -.  ( ( A  e.  B  /\  B  e.  A )  \/  A  e.  A )  <->  ( -.  ( A  e.  B  /\  B  e.  A
)  /\  -.  A  e.  A ) )
41, 2, 3sylanbrc 648 . 2  |-  ( Ord 
A  ->  -.  (
( A  e.  B  /\  B  e.  A
)  \/  A  e.  A ) )
5 elsuci 4430 . . . . 5  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
65anim2i 555 . . . 4  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( A  e.  B  /\  ( B  e.  A  \/  B  =  A ) ) )
7 andi 842 . . . 4  |-  ( ( A  e.  B  /\  ( B  e.  A  \/  B  =  A
) )  <->  ( ( A  e.  B  /\  B  e.  A )  \/  ( A  e.  B  /\  B  =  A
) ) )
86, 7sylib 190 . . 3  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( ( A  e.  B  /\  B  e.  A )  \/  ( A  e.  B  /\  B  =  A )
) )
9 eleq2 2319 . . . . 5  |-  ( B  =  A  ->  ( A  e.  B  <->  A  e.  A ) )
109biimpac 474 . . . 4  |-  ( ( A  e.  B  /\  B  =  A )  ->  A  e.  A )
1110orim2i 506 . . 3  |-  ( ( ( A  e.  B  /\  B  e.  A
)  \/  ( A  e.  B  /\  B  =  A ) )  -> 
( ( A  e.  B  /\  B  e.  A )  \/  A  e.  A ) )
128, 11syl 17 . 2  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( ( A  e.  B  /\  B  e.  A )  \/  A  e.  A ) )
134, 12nsyl 115 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    \/ wo 359    /\ wa 360    = wceq 1619    e. wcel 1621   Ord word 4363   suc csuc 4366
This theorem is referenced by:  onnbtwn  4456  ordsucss  4581
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-14 1626  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239  ax-sep 4115  ax-nul 4123  ax-pr 4186
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 941  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-eu 2122  df-mo 2123  df-clab 2245  df-cleq 2251  df-clel 2254  df-nfc 2383  df-ne 2423  df-ral 2523  df-rex 2524  df-rab 2527  df-v 2765  df-sbc 2967  df-dif 3130  df-un 3132  df-in 3134  df-ss 3141  df-nul 3431  df-if 3540  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3802  df-br 3998  df-opab 4052  df-tr 4088  df-eprel 4277  df-fr 4324  df-we 4326  df-ord 4367  df-suc 4370
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