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Theorem bj-indsuc 13703
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 13702 . . 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 4377 . . . 4  |-  ( x  =  B  ->  suc  x  =  suc  B )
43eleq1d 2233 . . 3  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
54rspcv 2824 . 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 1342    e. wcel 2135   A.wral 2442   (/)c0 3407   suc csuc 4340  Ind wind 13701
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 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146
This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-v 2726  df-un 3118  df-sn 3579  df-suc 4346  df-bj-ind 13702
This theorem is referenced by:  bj-indint  13706  bj-peano2  13714  bj-inf2vnlem2  13746
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