Theorem bdfind 16842

Description: Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4726 for a nonconstructive proof of the general case. See findset 16841 for a proof when A is assumed to be a set. (Contributed by BJ, 22-Nov-2019.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
bdfind.bd	|- BOUNDED A

Assertion

Ref	Expression
bdfind	|- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om )

Distinct variable group:   x, A

Proof of Theorem bdfind

Step	Hyp	Ref	Expression
1		bdfind.bd	. . . 4 |- BOUNDED A
2		bj-omex 16838	. . . 4 |- om e. _V
3	1, 2	bdssex 16798	. . 3 |- ( A C_ om -> A e. _V )
4	3	3ad2ant1 1045	. 2 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A e. _V )
5		findset 16841	. 2 |- ( A e. _V -> ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om ) )
6	4, 5	mpcom 36	1 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A = om )

Syntax hints:   -> wi 4   /\ w3a 1005   = wceq 1398   e. wcel 2205   A.wral 2522   _Vcvv 2815   C_ wss 3214   (/)c0 3512   suc csuc 4491   omcom 4717  BOUNDED wbdc 16736

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 2207  ax-14 2208  ax-ext 2216  ax-nul 4241  ax-pr 4327  ax-un 4559  ax-bd0 16709  ax-bdan 16711  ax-bdor 16712  ax-bdex 16715  ax-bdeq 16716  ax-bdel 16717  ax-bdsb 16718  ax-bdsep 16780  ax-infvn 16837

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-sn 3700  df-pr 3701  df-uni 3920  df-int 3955  df-suc 4497  df-iom 4718  df-bdc 16737  df-bj-ind 16823

This theorem is referenced by: (None)