Theorem bj-indsuc 16644

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 16643	. . 3 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
2	1	simprbi 275	. 2 |- (Ind A -> A. x e. A suc x e. A )
3		suceq 4505	. . . 4 |- ( x = B -> suc x = suc B )
4	3	eleq1d 2300	. . 3 |- ( x = B -> ( suc x e. A <-> suc B e. A ) )
5	4	rspcv 2907	. 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 1398   e. wcel 2202   A.wral 2511   (/)c0 3496   suc csuc 4468  Ind wind 16642

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-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-v 2805  df-un 3205  df-sn 3679  df-suc 4474  df-bj-ind 16643

This theorem is referenced by:  bj-indint  16647  bj-peano2  16655  bj-inf2vnlem2  16687