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Theorem bj-inf2vnlem3 10097
Description: Lemma for bj-inf2vn 10099. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)
Hypotheses
Ref Expression
bj-inf2vnlem3.bd1  |- BOUNDED  A
bj-inf2vnlem3.bd2  |- BOUNDED  Z
Assertion
Ref Expression
bj-inf2vnlem3  |-  ( 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-inf2vnlem3
Dummy variables  z  t  u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 bj-inf2vnlem2 10096 . . 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 bj-inf2vnlem3.bd1 . . . . . 6  |- BOUNDED  A
32bdeli 9966 . . . . 5  |- BOUNDED  z  e.  A
4 bj-inf2vnlem3.bd2 . . . . . 6  |- BOUNDED  Z
54bdeli 9966 . . . . 5  |- BOUNDED  z  e.  Z
63, 5ax-bdim 9934 . . . 4  |- BOUNDED  ( z  e.  A  ->  z  e.  Z )
7 nfv 1421 . . . 4  |-  F/ z ( t  e.  A  ->  t  e.  Z )
8 nfv 1421 . . . 4  |-  F/ z ( u  e.  A  ->  u  e.  Z )
9 nfv 1421 . . . 4  |-  F/ u
( z  e.  A  ->  z  e.  Z )
10 nfv 1421 . . . 4  |-  F/ u
( t  e.  A  ->  t  e.  Z )
11 eleq1 2100 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  A  <->  t  e.  A ) )
12 eleq1 2100 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  Z  <->  t  e.  Z ) )
1311, 12imbi12d 223 . . . . 5  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( t  e.  A  ->  t  e.  Z ) ) )
1413biimpd 132 . . . 4  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  ->  ( t  e.  A  ->  t  e.  Z ) ) )
15 eleq1 2100 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  A  <->  u  e.  A ) )
16 eleq1 2100 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  Z  <->  u  e.  Z ) )
1715, 16imbi12d 223 . . . . 5  |-  ( z  =  u  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( u  e.  A  ->  u  e.  Z ) ) )
1817biimprd 147 . . . 4  |-  ( z  =  u  ->  (
( u  e.  A  ->  u  e.  Z )  ->  ( z  e.  A  ->  z  e.  Z ) ) )
196, 7, 8, 9, 10, 14, 18bdsetindis 10094 . . 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 ) )
201, 19syl6 29 . 2  |-  ( A. x  e.  A  (
x  =  (/)  \/  E. y  e.  A  x  =  suc  y )  -> 
(Ind  Z  ->  A. z
( z  e.  A  ->  z  e.  Z ) ) )
21 dfss2 2934 . 2  |-  ( A 
C_  Z  <->  A. z
( z  e.  A  ->  z  e.  Z ) )
2220, 21syl6ibr 151 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 629   A.wal 1241    = wceq 1243    e. wcel 1393   A.wral 2306   E.wrex 2307    C_ wss 2917   (/)c0 3224   suc csuc 4102  BOUNDED wbdc 9960  Ind wind 10050
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-bdim 9934  ax-bdsetind 10093
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-suc 4108  df-bdc 9961  df-bj-ind 10051
This theorem is referenced by:  bj-inf2vn  10099
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