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Theorem onssneli 4683
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 4681 . . . 4  |-  ( B  e.  A  ->  B  e.  On )
3 eloni 4583 . . . 4  |-  ( B  e.  On  ->  Ord  B )
4 ordirr 4591 . . . 4  |-  ( Ord 
B  ->  -.  B  e.  B )
52, 3, 43syl 19 . . 3  |-  ( B  e.  A  ->  -.  B  e.  B )
6 ssel 3334 . . . 4  |-  ( A 
C_  B  ->  ( B  e.  A  ->  B  e.  B ) )
76com12 29 . . 3  |-  ( B  e.  A  ->  ( A  C_  B  ->  B  e.  B ) )
85, 7mtod 170 . 2  |-  ( B  e.  A  ->  -.  A  C_  B )
98con2i 114 1  |-  ( A 
C_  B  ->  -.  B  e.  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1725    C_ wss 3312   Ord word 4572   Oncon0 4573
This theorem is referenced by:  onsucconi  26152
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-sbc 3154  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-uni 4008  df-br 4205  df-opab 4259  df-tr 4295  df-eprel 4486  df-po 4495  df-so 4496  df-fr 4533  df-we 4535  df-ord 4576  df-on 4577
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