Theorem bj-inf2vnlem4 13855

Description: Lemma for bj-inf2vn2 13857. (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 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		nfv 1516	. . . 4 |- F/ z ( t e. A -> t e. Z )
3		nfv 1516	. . . 4 |- F/ z ( u e. A -> u e. Z )
4		nfv 1516	. . . 4 |- F/ u ( z e. A -> z e. Z )
5		nfv 1516	. . . 4 |- F/ u ( t e. A -> t e. Z )
6		eleq1 2229	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
7		eleq1 2229	. . . . . 6 |- ( z = t -> ( z e. Z <-> t e. Z ) )
8	6, 7	imbi12d 233	. . . . 5 |- ( z = t -> ( ( z e. A -> z e. Z ) <-> ( t e. A -> t e. Z ) ) )
9	8	biimpd 143	. . . 4 |- ( z = t -> ( ( z e. A -> z e. Z ) -> ( t e. A -> t e. Z ) ) )
10		eleq1 2229	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
11		eleq1 2229	. . . . . 6 |- ( z = u -> ( z e. Z <-> u e. Z ) )
12	10, 11	imbi12d 233	. . . . 5 |- ( z = u -> ( ( z e. A -> z e. Z ) <-> ( u e. A -> u e. Z ) ) )
13	12	biimprd 157	. . . 4 |- ( z = u -> ( ( u e. A -> u e. Z ) -> ( z e. A -> z e. Z ) ) )
14	2, 3, 4, 5, 9, 13	setindis 13849	. . 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 3131	. 2 |- ( A C_ Z <-> A. z ( z e. A -> z e. Z ) )
17	15, 16	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  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-setind 4514

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-bj-ind 13809

This theorem is referenced by:  bj-inf2vn2  13857