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Theorem bj-inf2vnlem4 10485
Description: Lemma for bj-inf2vn2 10487. (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 10483 . . 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 1437 . . . 4  |-  F/ z ( t  e.  A  ->  t  e.  Z )
3 nfv 1437 . . . 4  |-  F/ z ( u  e.  A  ->  u  e.  Z )
4 nfv 1437 . . . 4  |-  F/ u
( z  e.  A  ->  z  e.  Z )
5 nfv 1437 . . . 4  |-  F/ u
( t  e.  A  ->  t  e.  Z )
6 eleq1 2116 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  A  <->  t  e.  A ) )
7 eleq1 2116 . . . . . 6  |-  ( z  =  t  ->  (
z  e.  Z  <->  t  e.  Z ) )
86, 7imbi12d 227 . . . . 5  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( t  e.  A  ->  t  e.  Z ) ) )
98biimpd 136 . . . 4  |-  ( z  =  t  ->  (
( z  e.  A  ->  z  e.  Z )  ->  ( t  e.  A  ->  t  e.  Z ) ) )
10 eleq1 2116 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  A  <->  u  e.  A ) )
11 eleq1 2116 . . . . . 6  |-  ( z  =  u  ->  (
z  e.  Z  <->  u  e.  Z ) )
1210, 11imbi12d 227 . . . . 5  |-  ( z  =  u  ->  (
( z  e.  A  ->  z  e.  Z )  <-> 
( u  e.  A  ->  u  e.  Z ) ) )
1312biimprd 151 . . . 4  |-  ( z  =  u  ->  (
( u  e.  A  ->  u  e.  Z )  ->  ( z  e.  A  ->  z  e.  Z ) ) )
142, 3, 4, 5, 9, 13setindis 10479 . . 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 2962 . 2  |-  ( A 
C_  Z  <->  A. z
( z  e.  A  ->  z  e.  Z ) )
1715, 16syl6ibr 155 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 639   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323   E.wrex 2324    C_ wss 2945   (/)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-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-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-setind 4290
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-un 2950  df-in 2952  df-ss 2959  df-sn 3409  df-suc 4136  df-bj-ind 10438
This theorem is referenced by:  bj-inf2vn2  10487
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