Theorem bj-indsuc 13963

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 13962	. . 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 4387	. . . 4 |- ( x = B -> suc x = suc B )
4	3	eleq1d 2239	. . 3 |- ( x = B -> ( suc x e. A <-> suc B e. A ) )
5	4	rspcv 2830	. 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 1348   e. wcel 2141   A.wral 2448   (/)c0 3414   suc csuc 4350  Ind wind 13961

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-v 2732  df-un 3125  df-sn 3589  df-suc 4356  df-bj-ind 13962

This theorem is referenced by:  bj-indint  13966  bj-peano2  13974  bj-inf2vnlem2  14006