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Theorem trsuc 5526
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 5519 . . . . . 6  |-  B  C_  suc  B
2 ssexg 4570 . . . . . 6  |-  ( ( B  C_  suc  B  /\  suc  B  e.  A )  ->  B  e.  _V )
31, 2mpan 674 . . . . 5  |-  ( suc 
B  e.  A  ->  B  e.  _V )
4 sucidg 5520 . . . . 5  |-  ( B  e.  _V  ->  B  e.  suc  B )
53, 4syl 17 . . . 4  |-  ( suc 
B  e.  A  ->  B  e.  suc  B )
65ancri 554 . . 3  |-  ( suc 
B  e.  A  -> 
( B  e.  suc  B  /\  suc  B  e.  A ) )
7 trel 4525 . . 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 430 1  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 370    e. wcel 1872   _Vcvv 3080    C_ wss 3436   Tr wtr 4518   suc csuc 5444
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2057  ax-ext 2401  ax-sep 4546
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-clab 2408  df-cleq 2414  df-clel 2417  df-nfc 2568  df-v 3082  df-un 3441  df-in 3443  df-ss 3450  df-sn 3999  df-uni 4220  df-tr 4519  df-suc 5448
This theorem is referenced by:  onuninsuci  6682  limsuc  6691  tz7.44-2  7137  cantnflt  8186  cantnfp1lem3  8194  cantnflem1b  8200  cantnflem1  8203  cnfcom  8214  axdc3lem2  8889  inar1  9208  bnj967  29765  limsuc2  35870
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