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Theorem ordsuccl3 25207
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 4418 . . . 4  |-  ( B  e.  On  ->  Ord  B )
2 ordtr 4422 . . . 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 25206 . 2  |-  ( ( B  e.  On  /\  suc  A  e.  B )  ->  A  e.  B
)
6 trss 4138 . 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 1696    C_ wss 3165   Tr wtr 4129   Ord word 4407   Oncon0 4408   suc csuc 4410
This theorem is referenced by:  tartarmap  25991
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ral 2561  df-rex 2562  df-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-uni 3844  df-tr 4130  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-suc 4414
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