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Theorem onssneli 4823
Description: An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
onssneli  |-  ( A 
C_  B  ->  -.  B  e.  A )

Proof of Theorem onssneli
StepHypRef Expression
1 on.1 . . . . 5  |-  A  e.  On
21oneli 4821 . . . 4  |-  ( B  e.  A  ->  B  e.  On )
3 eloni 4724 . . . 4  |-  ( B  e.  On  ->  Ord  B )
4 ordirr 4732 . . . 4  |-  ( Ord 
B  ->  -.  B  e.  B )
52, 3, 43syl 20 . . 3  |-  ( B  e.  A  ->  -.  B  e.  B )
6 ssel 3345 . . . 4  |-  ( A 
C_  B  ->  ( B  e.  A  ->  B  e.  B ) )
76com12 31 . . 3  |-  ( B  e.  A  ->  ( A  C_  B  ->  B  e.  B ) )
85, 7mtod 177 . 2  |-  ( B  e.  A  ->  -.  A  C_  B )
98con2i 120 1  |-  ( A 
C_  B  ->  -.  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1756    C_ wss 3323   Ord word 4713   Oncon0 4714
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pr 4526
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2715  df-rex 2716  df-rab 2719  df-v 2969  df-sbc 3182  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-tr 4381  df-eprel 4627  df-po 4636  df-so 4637  df-fr 4674  df-we 4676  df-ord 4717  df-on 4718
This theorem is referenced by:  onsucconi  28235
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