MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ordnbtwn Unicode version

Theorem ordnbtwn 4663
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 4593 . . 3  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  A )
)
2 ordirr 4591 . . 3  |-  ( Ord 
A  ->  -.  A  e.  A )
3 ioran 477 . . 3  |-  ( -.  ( ( A  e.  B  /\  B  e.  A )  \/  A  e.  A )  <->  ( -.  ( A  e.  B  /\  B  e.  A
)  /\  -.  A  e.  A ) )
41, 2, 3sylanbrc 646 . 2  |-  ( Ord 
A  ->  -.  (
( A  e.  B  /\  B  e.  A
)  \/  A  e.  A ) )
5 elsuci 4639 . . . . 5  |-  ( B  e.  suc  A  -> 
( B  e.  A  \/  B  =  A
) )
65anim2i 553 . . . 4  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( A  e.  B  /\  ( B  e.  A  \/  B  =  A ) ) )
7 andi 838 . . . 4  |-  ( ( A  e.  B  /\  ( B  e.  A  \/  B  =  A
) )  <->  ( ( A  e.  B  /\  B  e.  A )  \/  ( A  e.  B  /\  B  =  A
) ) )
86, 7sylib 189 . . 3  |-  ( ( A  e.  B  /\  B  e.  suc  A )  ->  ( ( A  e.  B  /\  B  e.  A )  \/  ( A  e.  B  /\  B  =  A )
) )
9 eleq2 2496 . . . . 5  |-  ( B  =  A  ->  ( A  e.  B  <->  A  e.  A ) )
109biimpac 473 . . . 4  |-  ( ( A  e.  B  /\  B  =  A )  ->  A  e.  A )
1110orim2i 505 . . 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 115 1  |-  ( Ord 
A  ->  -.  ( A  e.  B  /\  B  e.  suc  A ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 358    /\ wa 359    = wceq 1652    e. wcel 1725   Ord word 4572   suc csuc 4575
This theorem is referenced by:  onnbtwn  4664  ordsucss  4789
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-fr 4533  df-we 4535  df-ord 4576  df-suc 4579
  Copyright terms: Public domain W3C validator