Theorem bdfind 15676

Description: Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4636 for a nonconstructive proof of the general case. See findset 15675 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 15672	. . . 4 |- om e. _V
3	1, 2	bdssex 15632	. . 3 |- ( A C_ om -> A e. _V )
4	3	3ad2ant1 1020	. 2 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A e. _V )
5		findset 15675	. 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 980   = wceq 1364   e. wcel 2167   A.wral 2475   _Vcvv 2763   C_ wss 3157   (/)c0 3451   suc csuc 4401   omcom 4627  BOUNDED wbdc 15570

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 4160  ax-pr 4243  ax-un 4469  ax-bd0 15543  ax-bdan 15545  ax-bdor 15546  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614  ax-infvn 15671

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 3452  df-sn 3629  df-pr 3630  df-uni 3841  df-int 3876  df-suc 4407  df-iom 4628  df-bdc 15571  df-bj-ind 15657

This theorem is referenced by: (None)