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Theorem onssnel2i 5522
Description: An ordering law for ordinal numbers. (Contributed by NM, 13-Jun-1994.)
Hypothesis
Ref Expression
on.1  |-  A  e.  On
Assertion
Ref Expression
onssnel2i  |-  ( B 
C_  A  ->  -.  A  e.  B )

Proof of Theorem onssnel2i
StepHypRef Expression
1 on.1 . . 3  |-  A  e.  On
21onirri 5518 . 2  |-  -.  A  e.  A
3 ssel 3438 . 2  |-  ( B 
C_  A  ->  ( A  e.  B  ->  A  e.  A ) )
42, 3mtoi 180 1  |-  ( B 
C_  A  ->  -.  A  e.  B )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    e. wcel 1844    C_ wss 3416   Oncon0 5412
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-9 1848  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-sep 4519  ax-nul 4527  ax-pr 4632
This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-3an 978  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-eu 2244  df-mo 2245  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3063  df-sbc 3280  df-dif 3419  df-un 3421  df-in 3423  df-ss 3430  df-nul 3741  df-if 3888  df-sn 3975  df-pr 3977  df-op 3981  df-uni 4194  df-br 4398  df-opab 4456  df-tr 4492  df-eprel 4736  df-po 4746  df-so 4747  df-fr 4784  df-we 4786  df-ord 5415  df-on 5416
This theorem is referenced by: (None)
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