Theorem bdfind 15882

Description: Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4647 for a nonconstructive proof of the general case. See findset 15881 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 15878	. . . 4 |- om e. _V
3	1, 2	bdssex 15838	. . 3 |- ( A C_ om -> A e. _V )
4	3	3ad2ant1 1021	. 2 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A e. _V )
5		findset 15881	. 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 981   = wceq 1373   e. wcel 2176   A.wral 2484   _Vcvv 2772   C_ wss 3166   (/)c0 3460   suc csuc 4412   omcom 4638  BOUNDED wbdc 15776

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-nul 4170  ax-pr 4253  ax-un 4480  ax-bd0 15749  ax-bdan 15751  ax-bdor 15752  ax-bdex 15755  ax-bdeq 15756  ax-bdel 15757  ax-bdsb 15758  ax-bdsep 15820  ax-infvn 15877

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-rab 2493  df-v 2774  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-sn 3639  df-pr 3640  df-uni 3851  df-int 3886  df-suc 4418  df-iom 4639  df-bdc 15777  df-bj-ind 15863

This theorem is referenced by: (None)