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Theorem trsuc 4666
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 4658 . . . . . 6  |-  B  C_  suc  B
2 ssexg 4349 . . . . . 6  |-  ( ( B  C_  suc  B  /\  suc  B  e.  A )  ->  B  e.  _V )
31, 2mpan 652 . . . . 5  |-  ( suc 
B  e.  A  ->  B  e.  _V )
4 sucidg 4659 . . . . 5  |-  ( B  e.  _V  ->  B  e.  suc  B )
53, 4syl 16 . . . 4  |-  ( suc 
B  e.  A  ->  B  e.  suc  B )
65ancri 536 . . 3  |-  ( suc 
B  e.  A  -> 
( B  e.  suc  B  /\  suc  B  e.  A ) )
7 trel 4309 . . 3  |-  ( Tr  A  ->  ( ( B  e.  suc  B  /\  suc  B  e.  A )  ->  B  e.  A
) )
86, 7syl5 30 . 2  |-  ( Tr  A  ->  ( suc  B  e.  A  ->  B  e.  A ) )
98imp 419 1  |-  ( ( Tr  A  /\  suc  B  e.  A )  ->  B  e.  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359    e. wcel 1725   _Vcvv 2956    C_ wss 3320   Tr wtr 4302   suc csuc 4583
This theorem is referenced by:  onuninsuci  4820  limsuc  4829  tz7.44-2  6665  cantnflt  7627  cantnfp1lem3  7636  cantnflem1b  7642  cantnflem1  7645  cnfcom  7657  axdc3lem2  8331  inar1  8650  limsuc2  27115  bnj967  29316
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-un 3325  df-in 3327  df-ss 3334  df-sn 3820  df-uni 4016  df-tr 4303  df-suc 4587
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