Theorem bj-indsuc 13810

Description: A direct consequence of the definition of Ind. (Contributed by BJ, 30-Nov-2019.)

Assertion

Ref	Expression
bj-indsuc	|- (Ind A -> ( B e. A -> suc B e. A ) )

Proof of Theorem bj-indsuc

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-bj-ind 13809	. . 3 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
2	1	simprbi 273	. 2 |- (Ind A -> A. x e. A suc x e. A )
3		suceq 4380	. . . 4 |- ( x = B -> suc x = suc B )
4	3	eleq1d 2235	. . 3 |- ( x = B -> ( suc x e. A <-> suc B e. A ) )
5	4	rspcv 2826	. 2 |- ( B e. A -> ( A. x e. A suc x e. A -> suc B e. A ) )
6	2, 5	syl5com 29	1 |- (Ind A -> ( B e. A -> suc B e. A ) )

Syntax hints:   -> wi 4   = wceq 1343   e. wcel 2136   A.wral 2444   (/)c0 3409   suc csuc 4343  Ind wind 13808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-v 2728  df-un 3120  df-sn 3582  df-suc 4349  df-bj-ind 13809

This theorem is referenced by:  bj-indint  13813  bj-peano2  13821  bj-inf2vnlem2  13853