Theorem findset 15558

Description: Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4635 for a nonconstructive proof of the general case. See bdfind 15559 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 2480	. . . . . . . 8 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
4		alral 2542	. . . . . . . 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 15557	. . . 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 3200	. 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 1362   = wceq 1364   e. wcel 2167   A.wral 2475   C_ wss 3157   (/)c0 3450   suc csuc 4400   omcom 4626

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4159  ax-pr 4242  ax-un 4468  ax-bd0 15426  ax-bdan 15428  ax-bdor 15429  ax-bdex 15432  ax-bdeq 15433  ax-bdel 15434  ax-bdsb 15435  ax-bdsep 15497  ax-infvn 15554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-rab 2484  df-v 2765  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3451  df-sn 3628  df-pr 3629  df-uni 3840  df-int 3875  df-suc 4406  df-iom 4627  df-bdc 15454  df-bj-ind 15540

This theorem is referenced by:  bdfind  15559