Theorem bdfind 14783

Description: Bounded induction (principle of induction when A is assumed to be bounded), proved from basic constructive axioms. See find 4600 for a nonconstructive proof of the general case. See findset 14782 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 14779	. . . 4 |- om e. _V
3	1, 2	bdssex 14739	. . 3 |- ( A C_ om -> A e. _V )
4	3	3ad2ant1 1018	. 2 |- ( ( A C_ om /\ (/) e. A /\ A. x e. A suc x e. A ) -> A e. _V )
5		findset 14782	. 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 978   = wceq 1353   e. wcel 2148   A.wral 2455   _Vcvv 2739   C_ wss 3131   (/)c0 3424   suc csuc 4367   omcom 4591  BOUNDED wbdc 14677

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-nul 4131  ax-pr 4211  ax-un 4435  ax-bd0 14650  ax-bdan 14652  ax-bdor 14653  ax-bdex 14656  ax-bdeq 14657  ax-bdel 14658  ax-bdsb 14659  ax-bdsep 14721  ax-infvn 14778

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-nul 3425  df-sn 3600  df-pr 3601  df-uni 3812  df-int 3847  df-suc 4373  df-iom 4592  df-bdc 14678  df-bj-ind 14764

This theorem is referenced by: (None)