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Theorem ordnbtwn 4973
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

Proof of Theorem ordnbtwn
StepHypRef Expression
1 ordn2lp 4903 . . 3
2 ordirr 4901 . . 3
3 ioran 490 . . 3
41, 2, 3sylanbrc 664 . 2
5 elsuci 4949 . . . . 5
65anim2i 569 . . . 4
7 andi 867 . . . 4
86, 7sylib 196 . . 3
9 eleq2 2530 . . . . 5
109biimpac 486 . . . 4
1110orim2i 518 . . 3
128, 11syl 16 . 2
134, 12nsyl 121 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  \/wo 368  /\wa 369  =wceq 1395  e.wcel 1818  Ordword 4882  succsuc 4885
This theorem is referenced by:  onnbtwn  4974  ordsucss  6653
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-fr 4843  df-we 4845  df-ord 4886  df-suc 4889
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