Theorem findset 10070

Description: Bounded induction (principle of induction when A is assumed to be a set) allowing a proof from basic constructive axioms. See find 4322 for a nonconstructive proof of the general case. See bdfind 10071 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 910	. . 3 |- ( ( A e. V /\ ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) ) -> A C_ om )
2		simp2 905	. . . . . 6 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> (/) e. A )
3		df-ral 2311	. . . . . . . 8 |- ( A. x e. A suc x e. A <-> A. x ( x e. A -> suc x e. A ) )
4		alral 2367	. . . . . . . 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 114	. . . . . . 7 |- ( A. x e. A suc x e. A -> A. x e. om ( x e. A -> suc x e. A ) )
6	5	3ad2ant3 927	. . . . . 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 290	. . . . 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 889	. . . . . 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 124	. . . . 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 270	. . . 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 10069	. . . 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 2962	. 2 |- ( ( A e. V /\ ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) ) -> A = om )
14	13	ex 108	1 |- ( A e. V -> ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om ) )

Syntax hints:   -> wi 4   /\ wa 97   /\ w3a 885   A.wal 1241   = wceq 1243   e. wcel 1393   A.wral 2306   C_ wss 2917   (/)c0 3224   suc csuc 4102   omcom 4313

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-nul 3883  ax-pr 3944  ax-un 4170  ax-bd0 9933  ax-bdan 9935  ax-bdor 9936  ax-bdex 9939  ax-bdeq 9940  ax-bdel 9941  ax-bdsb 9942  ax-bdsep 10004  ax-infvn 10066

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-pr 3382  df-uni 3581  df-int 3616  df-suc 4108  df-iom 4314  df-bdc 9961  df-bj-ind 10051

This theorem is referenced by:  bdfind  10071