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Theorem onssneli 4987
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 4985 . . . 4  |-  ( B  e.  A  ->  B  e.  On )
3 eloni 4888 . . . 4  |-  ( B  e.  On  ->  Ord  B )
4 ordirr 4896 . . . 4  |-  ( Ord 
B  ->  -.  B  e.  B )
52, 3, 43syl 20 . . 3  |-  ( B  e.  A  ->  -.  B  e.  B )
6 ssel 3498 . . . 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 1767    C_ wss 3476   Ord word 4877   Oncon0 4878
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882
This theorem is referenced by:  onsucconi  29479
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