Theorem bj-indsuc 13126

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 13125	. . 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 4324	. . . 4 |- ( x = B -> suc x = suc B )
4	3	eleq1d 2208	. . 3 |- ( x = B -> ( suc x e. A <-> suc B e. A ) )
5	4	rspcv 2785	. 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 1331   e. wcel 1480   A.wral 2416   (/)c0 3363   suc csuc 4287  Ind wind 13124

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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-suc 4293  df-bj-ind 13125

This theorem is referenced by:  bj-indint  13129  bj-peano2  13137  bj-inf2vnlem2  13169