Theorem findset 15745

Description: Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4645 for a nonconstructive proof of the general case. See bdfind 15746 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 1005	. . 3 |- ( ( A e. V /\ ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) ) -> A C_ om )
2		simp2 1000	. . . . . 6 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> (/) e. A )
3		df-ral 2488	. . . . . . . 8 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
4		alral 2550	. . . . . . . 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 1022	. . . . . 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 984	. . . . . 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 15744	. . . 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 3209	. 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 980   A.wal 1370   = wceq 1372   e. wcel 2175   A.wral 2483   C_ wss 3165   (/)c0 3459   suc csuc 4410   omcom 4636

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 615  ax-in2 616  ax-io 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-13 2177  ax-14 2178  ax-ext 2186  ax-nul 4169  ax-pr 4252  ax-un 4478  ax-bd0 15613  ax-bdan 15615  ax-bdor 15616  ax-bdex 15619  ax-bdeq 15620  ax-bdel 15621  ax-bdsb 15622  ax-bdsep 15684  ax-infvn 15741

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-ral 2488  df-rex 2489  df-rab 2492  df-v 2773  df-dif 3167  df-un 3169  df-in 3171  df-ss 3178  df-nul 3460  df-sn 3638  df-pr 3639  df-uni 3850  df-int 3885  df-suc 4416  df-iom 4637  df-bdc 15641  df-bj-ind 15727

This theorem is referenced by:  bdfind  15746