Theorem bdfind 17138

Description: Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4746 for a nonconstructive proof of the general case. See findset 17137 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 17134	. . . 4 |- om e. _V
3	1, 2	bdssex 17094	. . 3 |- ( A C_ om -> A e. _V )
4	3	3ad2ant1 1049	. 2 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A e. _V )
5		findset 17137	. 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 )

This proof depends on syntax axioms:   -> wi 4   /\ w3a 1009   = wceq 1402   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   (/)c0 3520   suc csuc 4510   omcom 4737  BOUNDED wbdc 17032

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-nul 4259  ax-pr 4346  ax-un 4578  ax-bd0 17005  ax-bdan 17007  ax-bdor 17008  ax-bdex 17011  ax-bdeq 17012  ax-bdel 17013  ax-bdsb 17014  ax-bdsep 17076  ax-infvn 17133

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3715  df-pr 3716  df-uni 3936  df-int 3971  df-suc 4516  df-iom 4738  df-bdc 17033  df-bj-ind 17119

This theorem is used by: (None)