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Theorem bj-indsuc 11480
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 11479 . . 3  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )
)
21simprbi 269 . 2  |-  (Ind  A  ->  A. x  e.  A  suc  x  e.  A )
3 suceq 4220 . . . 4  |-  ( x  =  B  ->  suc  x  =  suc  B )
43eleq1d 2156 . . 3  |-  ( x  =  B  ->  ( suc  x  e.  A  <->  suc  B  e.  A ) )
54rspcv 2718 . 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 1289    e. wcel 1438   A.wral 2359   (/)c0 3284   suc csuc 4183  Ind wind 11478
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-un 3001  df-sn 3447  df-suc 4189  df-bj-ind 11479
This theorem is referenced by:  bj-indint  11483  bj-peano2  11491  bj-inf2vnlem2  11523
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