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Theorem bj-indsuc 13115
Description: A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)
Assertion
Ref Expression
bj-indsuc  |-  (Ind  A  ->  ( B  e.  A  ->  suc  B  e.  A
) )

Proof of Theorem bj-indsuc
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 df-bj-ind 13114 . . 3  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
21simprbi 273 . 2  |-  (Ind  A  ->  A. x  e.  A  suc  x  e.  A )
3 suceq 4319 . . . 4  |-  ( x  =  B  ->  suc  x  =  suc  B )
43eleq1d 2206 . . 3  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
54rspcv 2780 . 2  |-  ( B  e.  A  ->  ( A. x  e.  A  suc  x  e.  A  ->  suc  B  e.  A ) )
62, 5syl5com 29 1  |-  (Ind  A  ->  ( B  e.  A  ->  suc  B  e.  A
) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1331    e. wcel 1480   A.wral 2414   (/)c0 3358   suc csuc 4282  Ind wind 13113
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-un 3070  df-sn 3528  df-suc 4288  df-bj-ind 13114
This theorem is referenced by:  bj-indint  13118  bj-peano2  13126  bj-inf2vnlem2  13158
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