Theorem bj-inf2vnlem3 10097

Description: Lemma for bj-inf2vn 10099. (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
bj-inf2vnlem3.bd1	|- BOUNDED A
bj-inf2vnlem3.bd2	|- BOUNDED Z

Assertion

Ref	Expression
bj-inf2vnlem3	|- ( 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-inf2vnlem3

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		bj-inf2vnlem3.bd1	. . . . . 6 |- BOUNDED A
3	2	bdeli 9966	. . . . 5 |- BOUNDED z e. A
4		bj-inf2vnlem3.bd2	. . . . . 6 |- BOUNDED Z
5	4	bdeli 9966	. . . . 5 |- BOUNDED z e. Z
6	3, 5	ax-bdim 9934	. . . 4 |- BOUNDED ( z e. A -> z e. Z )
7		nfv 1421	. . . 4 |- F/ z ( t e. A -> t e. Z )
8		nfv 1421	. . . 4 |- F/ z ( u e. A -> u e. Z )
9		nfv 1421	. . . 4 |- F/ u ( z e. A -> z e. Z )
10		nfv 1421	. . . 4 |- F/ u ( t e. A -> t e. Z )
11		eleq1 2100	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
12		eleq1 2100	. . . . . 6 |- ( z = t -> ( z e. Z <-> t e. Z ) )
13	11, 12	imbi12d 223	. . . . 5 |- ( z = t -> ( ( z e. A -> z e. Z ) <-> ( t e. A -> t e. Z ) ) )
14	13	biimpd 132	. . . 4 |- ( z = t -> ( ( z e. A -> z e. Z ) -> ( t e. A -> t e. Z ) ) )
15		eleq1 2100	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
16		eleq1 2100	. . . . . 6 |- ( z = u -> ( z e. Z <-> u e. Z ) )
17	15, 16	imbi12d 223	. . . . 5 |- ( z = u -> ( ( z e. A -> z e. Z ) <-> ( u e. A -> u e. Z ) ) )
18	17	biimprd 147	. . . 4 |- ( z = u -> ( ( u e. A -> u e. Z ) -> ( z e. A -> z e. Z ) ) )
19	6, 7, 8, 9, 10, 14, 18	bdsetindis 10094	. . 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 ) )
20	1, 19	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 ) ) )
21		dfss2 2934	. 2 |- ( A C_ Z <-> A. z ( z e. A -> z e. Z ) )
22	20, 21	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  BOUNDED wbdc 9960  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-bdim 9934  ax-bdsetind 10093

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-bdc 9961  df-bj-ind 10051

This theorem is referenced by:  bj-inf2vn  10099