Theorem bj-inf2vnlem4 14008

Description: Lemma for bj-inf2vn2 14010. (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 14006	. . 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 1521	. . . 4 |- F/ z ( t e. A -> t e. Z )
3		nfv 1521	. . . 4 |- F/ z ( u e. A -> u e. Z )
4		nfv 1521	. . . 4 |- F/ u ( z e. A -> z e. Z )
5		nfv 1521	. . . 4 |- F/ u ( t e. A -> t e. Z )
6		eleq1 2233	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
7		eleq1 2233	. . . . . 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 2233	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
11		eleq1 2233	. . . . . 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 14002	. . 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 3136	. 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 703   A.wal 1346   = wceq 1348   e. wcel 2141   A.wral 2448   E.wrex 2449   C_ wss 3121   (/)c0 3414   suc csuc 4350  Ind wind 13961

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152  ax-setind 4521

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-sn 3589  df-suc 4356  df-bj-ind 13962

This theorem is referenced by:  bj-inf2vn2  14010