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Theorem bj-inf2vnlem1 13002
Description: Lemma for bj-inf2vn 13006. 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 bi2 129 . . . . 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 682 . . . . . 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 119 . . . . 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 108 . . . . . 6  |-  ( ( ( x  =  (/)  ->  x  e.  A )  /\  ( E. y  e.  A  x  =  suc  y  ->  x  e.  A ) )  -> 
( x  =  (/)  ->  x  e.  A ) )
5 eleq1 2178 . . . . . 6  |-  ( x  =  (/)  ->  ( x  e.  A  <->  (/)  e.  A
) )
64, 5mpbidi 150 . . . . 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 1414 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. x ( x  =  (/)  ->  (/)  e.  A ) )
9 exim 1561 . . 3  |-  ( A. x ( x  =  (/)  ->  (/)  e.  A )  ->  ( E. x  x  =  (/)  ->  E. x (/) 
e.  A ) )
10 0ex 4023 . . . . . 6  |-  (/)  e.  _V
1110isseti 2666 . . . . 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 12803 . . . 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 113 . . . . . 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 1414 . . . 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 2115 . . . . 5  |-  suc  z  =  suc  z
21 suceq 4292 . . . . . . 7  |-  ( y  =  z  ->  suc  y  =  suc  z )
2221eqeq2d 2127 . . . . . 6  |-  ( y  =  z  ->  ( suc  z  =  suc  y 
<->  suc  z  =  suc  z ) )
2322rspcev 2761 . . . . 5  |-  ( ( z  e.  A  /\  suc  z  =  suc  z )  ->  E. y  e.  A  suc  z  =  suc  y )
2420, 23mpan2 419 . . . 4  |-  ( z  e.  A  ->  E. y  e.  A  suc  z  =  suc  y )
25 vex 2661 . . . . . 6  |-  z  e. 
_V
2625bj-sucex 12955 . . . . 5  |-  suc  z  e.  _V
27 eqeq1 2122 . . . . . . 7  |-  ( x  =  suc  z  -> 
( x  =  suc  y 
<->  suc  z  =  suc  y ) )
2827rexbidv 2413 . . . . . 6  |-  ( x  =  suc  z  -> 
( E. y  e.  A  x  =  suc  y 
<->  E. y  e.  A  suc  z  =  suc  y ) )
29 eleq1 2178 . . . . . 6  |-  ( x  =  suc  z  -> 
( x  e.  A  <->  suc  z  e.  A ) )
3028, 29imbi12d 233 . . . . 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 2751 . . . 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 2479 . 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 12959 . 2  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. z  e.  A  suc  z  e.  A )
)
3516, 33, 34sylanbrc 411 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 103    <-> wb 104    \/ wo 680   A.wal 1312    = wceq 1314   E.wex 1451    e. wcel 1463   A.wral 2391   E.wrex 2392   (/)c0 3331   suc csuc 4255  Ind wind 12958
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-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-nul 4022  ax-pr 4099  ax-un 4323  ax-bd0 12845  ax-bdor 12848  ax-bdex 12851  ax-bdeq 12852  ax-bdel 12853  ax-bdsb 12854  ax-bdsep 12916
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ral 2396  df-rex 2397  df-v 2660  df-dif 3041  df-un 3043  df-nul 3332  df-sn 3501  df-pr 3502  df-uni 3705  df-suc 4261  df-bdc 12873  df-bj-ind 12959
This theorem is referenced by:  bj-inf2vn  13006  bj-inf2vn2  13007
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