Theorem bj-indsuc 10052

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 10051	. . 3 |- (Ind A <-> ( (/) e. A /\ A. x e. A suc x e. A ) )
2	1	simprbi 260	. 2 |- (Ind A -> A. x e. A suc x e. A )
3		suceq 4139	. . . 4 |- ( x = B -> suc x = suc B )
4	3	eleq1d 2106	. . 3 |- ( x = B -> ( suc x e. A <-> suc B e. A ) )
5	4	rspcv 2652	. 2 |- ( B e. A -> ( A. x e. A suc x e. A -> suc B e. A ) )
6	2, 5	syl5com 26	1 |- (Ind A -> ( B e. A -> suc B e. A ) )

Syntax hints:   -> wi 4   = wceq 1243   e. wcel 1393   A.wral 2306   (/)c0 3224   suc csuc 4102  Ind wind 10050

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022

This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-v 2559  df-un 2922  df-sn 3381  df-suc 4108  df-bj-ind 10051

This theorem is referenced by:  bj-indint  10055  bj-peano2  10063  bj-inf2vnlem2  10096