Theorem bj-inf2vnlem4 10098

Description: Lemma for bj-inf2vn2 10100. (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 10096	. . 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 1421	. . . 4 |- F/ z ( t e. A -> t e. Z )
3		nfv 1421	. . . 4 |- F/ z ( u e. A -> u e. Z )
4		nfv 1421	. . . 4 |- F/ u ( z e. A -> z e. Z )
5		nfv 1421	. . . 4 |- F/ u ( t e. A -> t e. Z )
6		eleq1 2100	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
7		eleq1 2100	. . . . . 6 |- ( z = t -> ( z e. Z <-> t e. Z ) )
8	6, 7	imbi12d 223	. . . . 5 |- ( z = t -> ( ( z e. A -> z e. Z ) <-> ( t e. A -> t e. Z ) ) )
9	8	biimpd 132	. . . 4 |- ( z = t -> ( ( z e. A -> z e. Z ) -> ( t e. A -> t e. Z ) ) )
10		eleq1 2100	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
11		eleq1 2100	. . . . . 6 |- ( z = u -> ( z e. Z <-> u e. Z ) )
12	10, 11	imbi12d 223	. . . . 5 |- ( z = u -> ( ( z e. A -> z e. Z ) <-> ( u e. A -> u e. Z ) ) )
13	12	biimprd 147	. . . 4 |- ( z = u -> ( ( u e. A -> u e. Z ) -> ( z e. A -> z e. Z ) ) )
14	2, 3, 4, 5, 9, 13	setindis 10092	. . 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 29	. 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 2934	. 2 |- ( A C_ Z <-> A. z ( z e. A -> z e. Z ) )
17	15, 16	syl6ibr 151	1 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )

Syntax hints:   -> wi 4   \/ wo 629   A.wal 1241   = wceq 1243   e. wcel 1393   A.wral 2306   E.wrex 2307   C_ wss 2917   (/)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  ax-setind 4262

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-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-sn 3381  df-suc 4108  df-bj-ind 10051

This theorem is referenced by:  bj-inf2vn2  10100