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Theorem findset 11497
Description: Bounded induction (principle of induction when  A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4404 for a nonconstructive proof of the general case. See bdfind 11498 for a proof when  A is assumed to be bounded. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)
Assertion
Ref Expression
findset  |-  ( A  e.  V  ->  (
( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  =  om )
)
Distinct variable group:    x, A
Allowed substitution hint:    V( x)

Proof of Theorem findset
StepHypRef Expression
1 simpr1 949 . . 3  |-  ( ( A  e.  V  /\  ( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )
)  ->  A  C_  om )
2 simp2 944 . . . . . 6  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  (/)  e.  A )
3 df-ral 2364 . . . . . . . 8  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
4 alral 2421 . . . . . . . 8  |-  ( A. x ( x  e.  A  ->  suc  x  e.  A )  ->  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )
53, 4sylbi 119 . . . . . . 7  |-  ( A. x  e.  A  suc  x  e.  A  ->  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) )
653ad2ant3 966 . . . . . 6  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A
) )
72, 6jca 300 . . . . 5  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A
) ) )
8 3anass 928 . . . . . 6  |-  ( ( A  e.  V  /\  (/) 
e.  A  /\  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) )  <->  ( A  e.  V  /\  ( (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) ) ) )
98biimpri 131 . . . . 5  |-  ( ( A  e.  V  /\  ( (/)  e.  A  /\  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) ) )  ->  ( A  e.  V  /\  (/) 
e.  A  /\  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) ) )
107, 9sylan2 280 . . . 4  |-  ( ( A  e.  V  /\  ( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )
)  ->  ( A  e.  V  /\  (/)  e.  A  /\  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A
) ) )
11 speano5 11496 . . . 4  |-  ( ( A  e.  V  /\  (/) 
e.  A  /\  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) )  ->  om  C_  A
)
1210, 11syl 14 . . 3  |-  ( ( A  e.  V  /\  ( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )
)  ->  om  C_  A
)
131, 12eqssd 3040 . 2  |-  ( ( A  e.  V  /\  ( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )
)  ->  A  =  om )
1413ex 113 1  |-  ( A  e.  V  ->  (
( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  A  =  om )
)
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 924   A.wal 1287    = wceq 1289    e. wcel 1438   A.wral 2359    C_ wss 2997   (/)c0 3284   suc csuc 4183   omcom 4395
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-in1 579  ax-in2 580  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-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3957  ax-pr 4027  ax-un 4251  ax-bd0 11361  ax-bdan 11363  ax-bdor 11364  ax-bdex 11367  ax-bdeq 11368  ax-bdel 11369  ax-bdsb 11370  ax-bdsep 11432  ax-infvn 11493
This theorem depends on definitions:  df-bi 115  df-3an 926  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-rab 2368  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-sn 3447  df-pr 3448  df-uni 3649  df-int 3684  df-suc 4189  df-iom 4396  df-bdc 11389  df-bj-ind 11479
This theorem is referenced by:  bdfind  11498
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