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Theorem bj-inf2vnlem1 15126
Description: Lemma for bj-inf2vn 15130. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vnlem1  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> Ind  A )
Distinct variable group:    x, A, y

Proof of Theorem bj-inf2vnlem1
Dummy variable  z is distinct from all other variables.
StepHypRef Expression
1 biimpr 130 . . . . 5  |-  ( ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
( ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  ->  x  e.  A ) )
2 jaob 711 . . . . . 6  |-  ( ( ( x  =  (/)  \/ 
E. y  e.  A  x  =  suc  y )  ->  x  e.  A
)  <->  ( ( x  =  (/)  ->  x  e.  A )  /\  ( E. y  e.  A  x  =  suc  y  ->  x  e.  A )
) )
32biimpi 120 . . . . 5  |-  ( ( ( x  =  (/)  \/ 
E. y  e.  A  x  =  suc  y )  ->  x  e.  A
)  ->  ( (
x  =  (/)  ->  x  e.  A )  /\  ( E. y  e.  A  x  =  suc  y  ->  x  e.  A )
) )
4 simpl 109 . . . . . 6  |-  ( ( ( x  =  (/)  ->  x  e.  A )  /\  ( E. y  e.  A  x  =  suc  y  ->  x  e.  A ) )  -> 
( x  =  (/)  ->  x  e.  A ) )
5 eleq1 2252 . . . . . 6  |-  ( x  =  (/)  ->  ( x  e.  A  <->  (/)  e.  A
) )
64, 5mpbidi 151 . . . . 5  |-  ( ( ( x  =  (/)  ->  x  e.  A )  /\  ( E. y  e.  A  x  =  suc  y  ->  x  e.  A ) )  -> 
( x  =  (/)  -> 
(/)  e.  A )
)
71, 3, 63syl 17 . . . 4  |-  ( ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
( x  =  (/)  -> 
(/)  e.  A )
)
87alimi 1466 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. x ( x  =  (/)  ->  (/)  e.  A ) )
9 exim 1610 . . 3  |-  ( A. x ( x  =  (/)  ->  (/)  e.  A )  ->  ( E. x  x  =  (/)  ->  E. x (/) 
e.  A ) )
10 0ex 4145 . . . . . 6  |-  (/)  e.  _V
1110isseti 2760 . . . . 5  |-  E. x  x  =  (/)
12 pm2.27 40 . . . . 5  |-  ( E. x  x  =  (/)  ->  ( ( E. x  x  =  (/)  ->  E. x (/) 
e.  A )  ->  E. x (/)  e.  A
) )
1311, 12ax-mp 5 . . . 4  |-  ( ( E. x  x  =  (/)  ->  E. x (/)  e.  A
)  ->  E. x (/) 
e.  A )
14 bj-ex 14918 . . . 4  |-  ( E. x (/)  e.  A  -> 
(/)  e.  A )
1513, 14syl 14 . . 3  |-  ( ( E. x  x  =  (/)  ->  E. x (/)  e.  A
)  ->  (/)  e.  A
)
168, 9, 153syl 17 . 2  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  (/) 
e.  A )
173simprd 114 . . . . . 6  |-  ( ( ( x  =  (/)  \/ 
E. y  e.  A  x  =  suc  y )  ->  x  e.  A
)  ->  ( E. y  e.  A  x  =  suc  y  ->  x  e.  A ) )
181, 17syl 14 . . . . 5  |-  ( ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
( E. y  e.  A  x  =  suc  y  ->  x  e.  A
) )
1918alimi 1466 . . . 4  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. x ( E. y  e.  A  x  =  suc  y  ->  x  e.  A ) )
20 eqid 2189 . . . . 5  |-  suc  z  =  suc  z
21 suceq 4417 . . . . . . 7  |-  ( y  =  z  ->  suc  y  =  suc  z )
2221eqeq2d 2201 . . . . . 6  |-  ( y  =  z  ->  ( suc  z  =  suc  y 
<->  suc  z  =  suc  z ) )
2322rspcev 2856 . . . . 5  |-  ( ( z  e.  A  /\  suc  z  =  suc  z )  ->  E. y  e.  A  suc  z  =  suc  y )
2420, 23mpan2 425 . . . 4  |-  ( z  e.  A  ->  E. y  e.  A  suc  z  =  suc  y )
25 vex 2755 . . . . . 6  |-  z  e. 
_V
2625bj-sucex 15079 . . . . 5  |-  suc  z  e.  _V
27 eqeq1 2196 . . . . . . 7  |-  ( x  =  suc  z  -> 
( x  =  suc  y 
<->  suc  z  =  suc  y ) )
2827rexbidv 2491 . . . . . 6  |-  ( x  =  suc  z  -> 
( E. y  e.  A  x  =  suc  y 
<->  E. y  e.  A  suc  z  =  suc  y ) )
29 eleq1 2252 . . . . . 6  |-  ( x  =  suc  z  -> 
( x  e.  A  <->  suc  z  e.  A ) )
3028, 29imbi12d 234 . . . . 5  |-  ( x  =  suc  z  -> 
( ( E. y  e.  A  x  =  suc  y  ->  x  e.  A )  <->  ( E. y  e.  A  suc  z  =  suc  y  ->  suc  z  e.  A
) ) )
3126, 30spcv 2846 . . . 4  |-  ( A. x ( E. y  e.  A  x  =  suc  y  ->  x  e.  A )  ->  ( E. y  e.  A  suc  z  =  suc  y  ->  suc  z  e.  A ) )
3219, 24, 31syl2im 38 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> 
( z  e.  A  ->  suc  z  e.  A
) )
3332ralrimiv 2562 . 2  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. z  e.  A  suc  z  e.  A
)
34 df-bj-ind 15083 . 2  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. z  e.  A  suc  z  e.  A )
)
3516, 33, 34sylanbrc 417 1  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  -> Ind  A )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709   A.wal 1362    = wceq 1364   E.wex 1503    e. wcel 2160   A.wral 2468   E.wrex 2469   (/)c0 3437   suc csuc 4380  Ind wind 15082
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-nul 4144  ax-pr 4224  ax-un 4448  ax-bd0 14969  ax-bdor 14972  ax-bdex 14975  ax-bdeq 14976  ax-bdel 14977  ax-bdsb 14978  ax-bdsep 15040
This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-dif 3146  df-un 3148  df-nul 3438  df-sn 3613  df-pr 3614  df-uni 3825  df-suc 4386  df-bdc 14997  df-bj-ind 15083
This theorem is referenced by:  bj-inf2vn  15130  bj-inf2vn2  15131
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