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Theorem trsuc 5530
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 5523 . . . . . 6  |-  B  C_  suc  B
2 ssexg 4565 . . . . . 6  |-  ( ( B  C_  suc  B  /\  suc  B  e.  A )  ->  B  e.  _V )
31, 2mpan 681 . . . . 5  |-  ( suc 
B  e.  A  ->  B  e.  _V )
4 sucidg 5524 . . . . 5  |-  ( B  e.  _V  ->  B  e.  suc  B )
53, 4syl 17 . . . 4  |-  ( suc 
B  e.  A  ->  B  e.  suc  B )
65ancri 559 . . 3  |-  ( suc 
B  e.  A  -> 
( B  e.  suc  B  /\  suc  B  e.  A ) )
7 trel 4520 . . 3  |-  ( Tr  A  ->  ( ( B  e.  suc  B  /\  suc  B  e.  A )  ->  B  e.  A
) )
86, 7syl5 33 . 2  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
98imp 435 1  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 375    e. wcel 1898   _Vcvv 3057    C_ wss 3416   Tr wtr 4513   suc csuc 5448
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1680  ax-4 1693  ax-5 1769  ax-6 1816  ax-7 1862  ax-10 1926  ax-11 1931  ax-12 1944  ax-13 2102  ax-ext 2442  ax-sep 4541
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-tru 1458  df-ex 1675  df-nf 1679  df-sb 1809  df-clab 2449  df-cleq 2455  df-clel 2458  df-nfc 2592  df-v 3059  df-un 3421  df-in 3423  df-ss 3430  df-sn 3981  df-uni 4213  df-tr 4514  df-suc 5452
This theorem is referenced by:  onuninsuci  6699  limsuc  6708  tz7.44-2  7156  cantnflt  8208  cantnfp1lem3  8216  cantnflem1b  8222  cantnflem1  8225  cnfcom  8236  axdc3lem2  8912  inar1  9231  bnj967  29806  limsuc2  35945
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