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Theorem bj-inf2vnlem4 11868
Description: Lemma for bj-inf2vn2 11870. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
bj-inf2vnlem4  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A  C_  Z ) )
Distinct variable groups:    x, y, A   
x, Z, y

Proof of Theorem bj-inf2vnlem4
Dummy variables  z  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem2 11866 . . 3  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A. u
( A. t  e.  u  ( t  e.  A  ->  t  e.  Z )  ->  (
u  e.  A  ->  u  e.  Z )
) ) )
2 nfv 1466 . . . 4  |-  F/ z ( t  e.  A  ->  t  e.  Z )
3 nfv 1466 . . . 4  |-  F/ z ( u  e.  A  ->  u  e.  Z )
4 nfv 1466 . . . 4  |-  F/ u
( z  e.  A  ->  z  e.  Z )
5 nfv 1466 . . . 4  |-  F/ u
( t  e.  A  ->  t  e.  Z )
6 eleq1 2150 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  A  <->  t  e.  A ) )
7 eleq1 2150 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  Z  <->  t  e.  Z ) )
86, 7imbi12d 232 . . . . 5  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( t  e.  A  ->  t  e.  Z ) ) )
98biimpd 142 . . . 4  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  ->  ( t  e.  A  ->  t  e.  Z ) ) )
10 eleq1 2150 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  A  <->  u  e.  A ) )
11 eleq1 2150 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  Z  <->  u  e.  Z ) )
1210, 11imbi12d 232 . . . . 5  |-  ( z  =  u  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( u  e.  A  ->  u  e.  Z ) ) )
1312biimprd 156 . . . 4  |-  ( z  =  u  ->  (
( u  e.  A  ->  u  e.  Z )  ->  ( z  e.  A  ->  z  e.  Z ) ) )
142, 3, 4, 5, 9, 13setindis 11862 . . 3  |-  ( A. u ( A. t  e.  u  ( t  e.  A  ->  t  e.  Z )  ->  (
u  e.  A  ->  u  e.  Z )
)  ->  A. z
( z  e.  A  ->  z  e.  Z ) )
151, 14syl6 33 . 2  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A. z
( z  e.  A  ->  z  e.  Z ) ) )
16 dfss2 3014 . 2  |-  ( A 
C_  Z  <->  A. z
( z  e.  A  ->  z  e.  Z ) )
1715, 16syl6ibr 160 1  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A  C_  Z ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    \/ wo 664   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359   E.wrex 2360    C_ wss 2999   (/)c0 3286   suc csuc 4192  Ind wind 11821
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  ax-setind 4353
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-rex 2365  df-v 2621  df-un 3003  df-in 3005  df-ss 3012  df-sn 3452  df-suc 4198  df-bj-ind 11822
This theorem is referenced by:  bj-inf2vn2  11870
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