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Theorem bj-inf2vnlem1 10482
Description: Lemma for bj-inf2vn 10486. 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 125 . . . . 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 641 . . . . . 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 117 . . . . 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 106 . . . . . 6  |-  ( ( ( x  =  (/)  ->  x  e.  A )  /\  ( E. y  e.  A  x  =  suc  y  ->  x  e.  A ) )  -> 
( x  =  (/)  ->  x  e.  A ) )
5 eleq1 2116 . . . . . 6  |-  ( x  =  (/)  ->  ( x  e.  A  <->  (/)  e.  A
) )
64, 5mpbidi 144 . . . . 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 1360 . . 3  |-  ( A. x ( x  e.  A  <->  ( x  =  (/)  \/  E. y  e.  A  x  =  suc  y ) )  ->  A. x ( x  =  (/)  ->  (/)  e.  A ) )
9 exim 1506 . . 3  |-  ( A. x ( x  =  (/)  ->  (/)  e.  A )  ->  ( E. x  x  =  (/)  ->  E. x (/) 
e.  A ) )
10 0ex 3912 . . . . . 6  |-  (/)  e.  _V
1110isseti 2580 . . . . 5  |-  E. x  x  =  (/)
12 pm2.27 39 . . . . 5  |-  ( E. x  x  =  (/)  ->  ( ( E. x  x  =  (/)  ->  E. x (/) 
e.  A )  ->  E. x (/)  e.  A
) )
1311, 12ax-mp 7 . . . 4  |-  ( ( E. x  x  =  (/)  ->  E. x (/)  e.  A
)  ->  E. x (/) 
e.  A )
14 bj-ex 10289 . . . 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 111 . . . . . 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 1360 . . . 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 2056 . . . . 5  |-  suc  z  =  suc  z
21 suceq 4167 . . . . . . 7  |-  ( y  =  z  ->  suc  y  =  suc  z )
2221eqeq2d 2067 . . . . . 6  |-  ( y  =  z  ->  ( suc  z  =  suc  y 
<->  suc  z  =  suc  z ) )
2322rspcev 2673 . . . . 5  |-  ( ( z  e.  A  /\  suc  z  =  suc  z )  ->  E. y  e.  A  suc  z  =  suc  y )
2420, 23mpan2 409 . . . 4  |-  ( z  e.  A  ->  E. y  e.  A  suc  z  =  suc  y )
25 vex 2577 . . . . . 6  |-  z  e. 
_V
2625bj-sucex 10430 . . . . 5  |-  suc  z  e.  _V
27 eqeq1 2062 . . . . . . 7  |-  ( x  =  suc  z  -> 
( x  =  suc  y 
<->  suc  z  =  suc  y ) )
2827rexbidv 2344 . . . . . 6  |-  ( x  =  suc  z  -> 
( E. y  e.  A  x  =  suc  y 
<->  E. y  e.  A  suc  z  =  suc  y ) )
29 eleq1 2116 . . . . . 6  |-  ( x  =  suc  z  -> 
( x  e.  A  <->  suc  z  e.  A ) )
3028, 29imbi12d 227 . . . . 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 2663 . . . 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 2408 . 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 10438 . 2  |-  (Ind  A  <->  (
(/)  e.  A  /\  A. z  e.  A  suc  z  e.  A )
)
3516, 33, 34sylanbrc 402 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 101    <-> wb 102    \/ wo 639   A.wal 1257    = wceq 1259   E.wex 1397    e. wcel 1409   A.wral 2323   E.wrex 2324   (/)c0 3252   suc csuc 4130  Ind wind 10437
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911  ax-pr 3972  ax-un 4198  ax-bd0 10320  ax-bdor 10323  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2948  df-un 2950  df-nul 3253  df-sn 3409  df-pr 3410  df-uni 3609  df-suc 4136  df-bdc 10348  df-bj-ind 10438
This theorem is referenced by:  bj-inf2vn  10486  bj-inf2vn2  10487
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