Theorem bj-inf2vnlem3 13854

Description: Lemma for bj-inf2vn 13856. (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 13853	. . 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 13728	. . . . 5 |- BOUNDED z e. A
4		bj-inf2vnlem3.bd2	. . . . . 6 |- BOUNDED Z
5	4	bdeli 13728	. . . . 5 |- BOUNDED z e. Z
6	3, 5	ax-bdim 13696	. . . 4 |- BOUNDED ( z e. A -> z e. Z )
7		nfv 1516	. . . 4 |- F/ z ( t e. A -> t e. Z )
8		nfv 1516	. . . 4 |- F/ z ( u e. A -> u e. Z )
9		nfv 1516	. . . 4 |- F/ u ( z e. A -> z e. Z )
10		nfv 1516	. . . 4 |- F/ u ( t e. A -> t e. Z )
11		eleq1 2229	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
12		eleq1 2229	. . . . . 6 |- ( z = t -> ( z e. Z <-> t e. Z ) )
13	11, 12	imbi12d 233	. . . . 5 |- ( z = t -> ( ( z e. A -> z e. Z ) <-> ( t e. A -> t e. Z ) ) )
14	13	biimpd 143	. . . 4 |- ( z = t -> ( ( z e. A -> z e. Z ) -> ( t e. A -> t e. Z ) ) )
15		eleq1 2229	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
16		eleq1 2229	. . . . . 6 |- ( z = u -> ( z e. Z <-> u e. Z ) )
17	15, 16	imbi12d 233	. . . . 5 |- ( z = u -> ( ( z e. A -> z e. Z ) <-> ( u e. A -> u e. Z ) ) )
18	17	biimprd 157	. . . 4 |- ( z = u -> ( ( u e. A -> u e. Z ) -> ( z e. A -> z e. Z ) ) )
19	6, 7, 8, 9, 10, 14, 18	bdsetindis 13851	. . 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 33	. 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 3131	. 2 |- ( A C_ Z <-> A. z ( z e. A -> z e. Z ) )
22	20, 21	syl6ibr 161	1 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )

Syntax hints:   -> wi 4   \/ wo 698   A.wal 1341   = wceq 1343   e. wcel 2136   A.wral 2444   E.wrex 2445   C_ wss 3116   (/)c0 3409   suc csuc 4343  BOUNDED wbdc 13722  Ind wind 13808

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147  ax-bdim 13696  ax-bdsetind 13850

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-sn 3582  df-suc 4349  df-bdc 13723  df-bj-ind 13809

This theorem is referenced by:  bj-inf2vn  13856