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Theorem trsuc 4492
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 4485 . . . . . 6  |-  B  C_  suc  B
2 ssexg 4176 . . . . . 6  |-  ( ( B  C_  suc  B  /\  suc  B  e.  A )  ->  B  e.  _V )
31, 2mpan 651 . . . . 5  |-  ( suc 
B  e.  A  ->  B  e.  _V )
4 sucidg 4486 . . . . 5  |-  ( B  e.  _V  ->  B  e.  suc  B )
53, 4syl 15 . . . 4  |-  ( suc 
B  e.  A  ->  B  e.  suc  B )
65ancri 535 . . 3  |-  ( suc 
B  e.  A  -> 
( B  e.  suc  B  /\  suc  B  e.  A ) )
7 trel 4136 . . 3  |-  ( Tr  A  ->  ( ( B  e.  suc  B  /\  suc  B  e.  A )  ->  B  e.  A
) )
86, 7syl5 28 . 2  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
98imp 418 1  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 358    e. wcel 1696   _Vcvv 2801    C_ wss 3165   Tr wtr 4129   suc csuc 4410
This theorem is referenced by:  onuninsuci  4647  limsuc  4656  tz7.44-2  6436  cantnflt  7389  cantnfp1lem3  7398  cantnflem1b  7404  cantnflem1  7407  cnfcom  7419  axdc3lem2  8093  inar1  8413  ordsuccl  25205  ordsuccl2  25206  tartarmap  25991  limsuc2  27240  bnj967  29293
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-v 2803  df-un 3170  df-in 3172  df-ss 3179  df-sn 3659  df-uni 3844  df-tr 4130  df-suc 4414
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