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Theorem onssneli 4996
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 4994 . . . 4  |-  ( B  e.  A  ->  B  e.  On )
3 eloni 4897 . . . 4  |-  ( B  e.  On  ->  Ord  B )
4 ordirr 4905 . . . 4  |-  ( Ord 
B  ->  -.  B  e.  B )
52, 3, 43syl 20 . . 3  |-  ( B  e.  A  ->  -.  B  e.  B )
6 ssel 3493 . . . 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 1819    C_ wss 3471   Ord word 4886   Oncon0 4887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  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 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-tr 4551  df-eprel 4800  df-po 4809  df-so 4810  df-fr 4847  df-we 4849  df-ord 4890  df-on 4891
This theorem is referenced by:  onsucconi  30107
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