Theorem bj-inf2vnlem4 10768

Description: Lemma for bj-inf2vn2 10770. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inf2vnlem4	|- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )

Distinct variable groups:   x, y, A   x, Z, y

Proof of Theorem bj-inf2vnlem4

Dummy variables z t u are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bj-inf2vnlem2 10766	. . 3 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) ) )
2		nfv 1461	. . . 4 |- F/ z ( t e. A -> t e. Z )
3		nfv 1461	. . . 4 |- F/ z ( u e. A -> u e. Z )
4		nfv 1461	. . . 4 |- F/ u ( z e. A -> z e. Z )
5		nfv 1461	. . . 4 |- F/ u ( t e. A -> t e. Z )
6		eleq1 2141	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
7		eleq1 2141	. . . . . 6 |- ( z = t -> ( z e. Z <-> t e. Z ) )
8	6, 7	imbi12d 232	. . . . 5 |- ( z = t -> ( ( z e. A -> z e. Z ) <-> ( t e. A -> t e. Z ) ) )
9	8	biimpd 142	. . . 4 |- ( z = t -> ( ( z e. A -> z e. Z ) -> ( t e. A -> t e. Z ) ) )
10		eleq1 2141	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
11		eleq1 2141	. . . . . 6 |- ( z = u -> ( z e. Z <-> u e. Z ) )
12	10, 11	imbi12d 232	. . . . 5 |- ( z = u -> ( ( z e. A -> z e. Z ) <-> ( u e. A -> u e. Z ) ) )
13	12	biimprd 156	. . . 4 |- ( z = u -> ( ( u e. A -> u e. Z ) -> ( z e. A -> z e. Z ) ) )
14	2, 3, 4, 5, 9, 13	setindis 10762	. . 3 |- ( A. u ( A. t e. u ( t e. A -> t e. Z ) -> ( u e. A -> u e. Z ) ) -> A. z ( z e. A -> z e. Z ) )
15	1, 14	syl6 33	. 2 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A. z ( z e. A -> z e. Z ) ) )
16		dfss2 2988	. 2 |- ( A C_ Z <-> A. z ( z e. A -> z e. Z ) )
17	15, 16	syl6ibr 160	1 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )

Syntax hints:   -> wi 4   \/ wo 661   A.wal 1282   = wceq 1284   e. wcel 1433   A.wral 2348   E.wrex 2349   C_ wss 2973   (/)c0 3251   suc csuc 4120  Ind wind 10721

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-setind 4280

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-sn 3404  df-suc 4126  df-bj-ind 10722

This theorem is referenced by:  bj-inf2vn2  10770