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Theorem onssneli 5539
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 5537 . . . 4  |-  ( B  e.  A  ->  B  e.  On )
3 eloni 5440 . . . 4  |-  ( B  e.  On  ->  Ord  B )
4 ordirr 5448 . . . 4  |-  ( Ord 
B  ->  -.  B  e.  B )
52, 3, 43syl 18 . . 3  |-  ( B  e.  A  ->  -.  B  e.  B )
6 ssel 3412 . . . 4  |-  ( A 
C_  B  ->  ( B  e.  A  ->  B  e.  B ) )
76com12 31 . . 3  |-  ( B  e.  A  ->  ( A  C_  B  ->  B  e.  B ) )
85, 7mtod 182 . 2  |-  ( B  e.  A  ->  -.  A  C_  B )
98con2i 124 1  |-  ( A 
C_  B  ->  -.  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1904    C_ wss 3390   Ord word 5429   Oncon0 5430
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-tr 4491  df-eprel 4750  df-po 4760  df-so 4761  df-fr 4798  df-we 4800  df-ord 5433  df-on 5434
This theorem is referenced by:  onsucconi  31168
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