Theorem findset 16644

Description: Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4703 for a nonconstructive proof of the general case. See bdfind 16645 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

Step	Hyp	Ref	Expression
1		simpr1 1030	. . 3 |- ( ( A e. V /\ ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) ) -> A C_ om )
2		simp2 1025	. . . . . 6 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> (/) e. A )
3		df-ral 2516	. . . . . . . 8 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
4		alral 2578	. . . . . . . 8 |- ( A. x ( x e. A -> suc x e. A ) -> A. x e. om ( x e. A -> suc x e. A ) )
5	3, 4	sylbi 121	. . . . . . 7 |- ( A. x e. A suc x e. A -> A. x e. om ( x e. A -> suc x e. A ) )
6	5	3ad2ant3 1047	. . . . . 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 ) )
7	2, 6	jca 306	. . . . 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 1009	. . . . . 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 ) ) ) )
9	8	biimpri 133	. . . . 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 ) ) )
10	7, 9	sylan2 286	. . . 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 16643	. . . 4 |- ( ( A e. V /\ (/) e. A /\ A. x e. om ( x e. A -> suc x e. A ) ) -> om C_ A )
12	10, 11	syl 14	. . 3 |- ( ( A e. V /\ ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) ) -> om C_ A )
13	1, 12	eqssd 3245	. 2 |- ( ( A e. V /\ ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) ) -> A = om )
14	13	ex 115	1 |- ( A e. V -> ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om ) )

Syntax hints:   -> wi 4   /\ wa 104   /\ w3a 1005   A.wal 1396   = wceq 1398   e. wcel 2202   A.wral 2511   C_ wss 3201   (/)c0 3496   suc csuc 4468   omcom 4694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-nul 4220  ax-pr 4305  ax-un 4536  ax-bd0 16512  ax-bdan 16514  ax-bdor 16515  ax-bdex 16518  ax-bdeq 16519  ax-bdel 16520  ax-bdsb 16521  ax-bdsep 16583  ax-infvn 16640

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-rab 2520  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-sn 3679  df-pr 3680  df-uni 3899  df-int 3934  df-suc 4474  df-iom 4695  df-bdc 16540  df-bj-ind 16626

This theorem is referenced by:  bdfind  16645