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Theorem trsuc 4249
Description: A set whose successor belongs to a transitive class also belongs. (Contributed by NM, 5-Sep-2003.) (Proof shortened by Andrew Salmon, 12-Aug-2011.)
Assertion
Ref Expression
trsuc  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )

Proof of Theorem trsuc
StepHypRef Expression
1 sssucid 4242 . . . . . 6  |-  B  C_  suc  B
2 ssexg 3978 . . . . . 6  |-  ( ( B  C_  suc  B  /\  suc  B  e.  A )  ->  B  e.  _V )
31, 2mpan 415 . . . . 5  |-  ( suc 
B  e.  A  ->  B  e.  _V )
4 sucidg 4243 . . . . 5  |-  ( B  e.  _V  ->  B  e.  suc  B )
53, 4syl 14 . . . 4  |-  ( suc 
B  e.  A  ->  B  e.  suc  B )
65ancri 317 . . 3  |-  ( suc 
B  e.  A  -> 
( B  e.  suc  B  /\  suc  B  e.  A ) )
7 trel 3943 . . 3  |-  ( Tr  A  ->  ( ( B  e.  suc  B  /\  suc  B  e.  A )  ->  B  e.  A
) )
86, 7syl5 32 . 2  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
98imp 122 1  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    e. wcel 1438   _Vcvv 2619    C_ wss 2999   Tr wtr 3936   suc csuc 4192
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3957
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-uni 3654  df-tr 3937  df-suc 4198
This theorem is referenced by:  nnnninf  6806
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