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Theorem bj-inf2vnlem4 13098
Description: Lemma for bj-inf2vn2 13100. (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 13096 . . 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 1493 . . . 4  |-  F/ z ( t  e.  A  ->  t  e.  Z )
3 nfv 1493 . . . 4  |-  F/ z ( u  e.  A  ->  u  e.  Z )
4 nfv 1493 . . . 4  |-  F/ u
( z  e.  A  ->  z  e.  Z )
5 nfv 1493 . . . 4  |-  F/ u
( t  e.  A  ->  t  e.  Z )
6 eleq1 2180 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  A  <->  t  e.  A ) )
7 eleq1 2180 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  Z  <->  t  e.  Z ) )
86, 7imbi12d 233 . . . . 5  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( t  e.  A  ->  t  e.  Z ) ) )
98biimpd 143 . . . 4  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  ->  ( t  e.  A  ->  t  e.  Z ) ) )
10 eleq1 2180 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  A  <->  u  e.  A ) )
11 eleq1 2180 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  Z  <->  u  e.  Z ) )
1210, 11imbi12d 233 . . . . 5  |-  ( z  =  u  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( u  e.  A  ->  u  e.  Z ) ) )
1312biimprd 157 . . . 4  |-  ( z  =  u  ->  (
( u  e.  A  ->  u  e.  Z )  ->  ( z  e.  A  ->  z  e.  Z ) ) )
142, 3, 4, 5, 9, 13setindis 13092 . . 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 3056 . 2  |-  ( A 
C_  Z  <->  A. z
( z  e.  A  ->  z  e.  Z ) )
1715, 16syl6ibr 161 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 682   A.wal 1314    = wceq 1316    e. wcel 1465   A.wral 2393   E.wrex 2394    C_ wss 3041   (/)c0 3333   suc csuc 4257  Ind wind 13051
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-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-tru 1319  df-nf 1422  df-sb 1721  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-sn 3503  df-suc 4263  df-bj-ind 13052
This theorem is referenced by:  bj-inf2vn2  13100
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