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Theorem ordsuccl3 25104
Description: If a successor of  A belongs to an ordinal,  A is a part of the ordinal. (Contributed by FL, 20-Mar-2011.)
Assertion
Ref Expression
ordsuccl3  |-  ( ( B  e.  On  /\  suc  A  e.  B )  ->  A  C_  B
)

Proof of Theorem ordsuccl3
StepHypRef Expression
1 eloni 4402 . . . 4  |-  ( B  e.  On  ->  Ord  B )
2 ordtr 4406 . . . 4  |-  ( Ord 
B  ->  Tr  B
)
31, 2syl 15 . . 3  |-  ( B  e.  On  ->  Tr  B )
43adantr 451 . 2  |-  ( ( B  e.  On  /\  suc  A  e.  B )  ->  Tr  B )
5 ordsuccl2 25103 . 2  |-  ( ( B  e.  On  /\  suc  A  e.  B )  ->  A  e.  B
)
6 trss 4122 . 2  |-  ( Tr  B  ->  ( A  e.  B  ->  A  C_  B ) )
74, 5, 6sylc 56 1  |-  ( ( B  e.  On  /\  suc  A  e.  B )  ->  A  C_  B
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1684    C_ wss 3152   Tr wtr 4113   Ord word 4391   Oncon0 4392   suc csuc 4394
This theorem is referenced by:  tartarmap  25888
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ral 2548  df-rex 2549  df-v 2790  df-un 3157  df-in 3159  df-ss 3166  df-sn 3646  df-uni 3828  df-tr 4114  df-po 4314  df-so 4315  df-fr 4352  df-we 4354  df-ord 4395  df-on 4396  df-suc 4398
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