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Theorem findset 10457
Description: Bounded induction (principle of induction when  A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4350 for a nonconstructive proof of the general case. See bdfind 10458 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 921 . . 3  |-  ( ( A  e.  V  /\  ( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )
)  ->  A  C_  om )
2 simp2 916 . . . . . 6  |-  ( ( A  C_  om  /\  (/)  e.  A  /\  A. x  e.  A  suc  x  e.  A )  ->  (/)  e.  A )
3 df-ral 2328 . . . . . . . 8  |-  ( A. x  e.  A  suc  x  e.  A  <->  A. x
( x  e.  A  ->  suc  x  e.  A
) )
4 alral 2384 . . . . . . . 8  |-  ( A. x ( x  e.  A  ->  suc  x  e.  A )  ->  A. x  e.  om  ( x  e.  A  ->  suc  x  e.  A ) )
53, 4sylbi 118 . . . . . . 7  |-  ( A. x  e.  A  suc  x  e.  A  ->  A. x  e.  om  (
x  e.  A  ->  suc  x  e.  A ) )
653ad2ant3 938 . . . . . 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 294 . . . . 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 900 . . . . . 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 128 . . . . 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 274 . . . 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 10456 . . . 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 2990 . 2  |-  ( ( A  e.  V  /\  ( A  C_  om  /\  (/) 
e.  A  /\  A. x  e.  A  suc  x  e.  A )
)  ->  A  =  om )
1413ex 112 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 101    /\ w3a 896   A.wal 1257    = wceq 1259    e. wcel 1409   A.wral 2323    C_ wss 2945   (/)c0 3252   suc csuc 4130   omcom 4341
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-in1 554  ax-in2 555  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-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-nul 3911  ax-pr 3972  ax-un 4198  ax-bd0 10320  ax-bdan 10322  ax-bdor 10323  ax-bdex 10326  ax-bdeq 10327  ax-bdel 10328  ax-bdsb 10329  ax-bdsep 10391  ax-infvn 10453
This theorem depends on definitions:  df-bi 114  df-3an 898  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-rab 2332  df-v 2576  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-sn 3409  df-pr 3410  df-uni 3609  df-int 3644  df-suc 4136  df-iom 4342  df-bdc 10348  df-bj-ind 10438
This theorem is referenced by:  bdfind  10458
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