Theorem bj-inf2vnlem4 16108

Description: Lemma for bj-inf2vn2 16110. (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 16106	. . 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 1552	. . . 4 |- F/ z ( t e. A -> t e. Z )
3		nfv 1552	. . . 4 |- F/ z ( u e. A -> u e. Z )
4		nfv 1552	. . . 4 |- F/ u ( z e. A -> z e. Z )
5		nfv 1552	. . . 4 |- F/ u ( t e. A -> t e. Z )
6		eleq1 2270	. . . . . 6 |- ( z = t -> ( z e. A <-> t e. A ) )
7		eleq1 2270	. . . . . 6 |- ( z = t -> ( z e. Z <-> t e. Z ) )
8	6, 7	imbi12d 234	. . . . 5 |- ( z = t -> ( ( z e. A -> z e. Z ) <-> ( t e. A -> t e. Z ) ) )
9	8	biimpd 144	. . . 4 |- ( z = t -> ( ( z e. A -> z e. Z ) -> ( t e. A -> t e. Z ) ) )
10		eleq1 2270	. . . . . 6 |- ( z = u -> ( z e. A <-> u e. A ) )
11		eleq1 2270	. . . . . 6 |- ( z = u -> ( z e. Z <-> u e. Z ) )
12	10, 11	imbi12d 234	. . . . 5 |- ( z = u -> ( ( z e. A -> z e. Z ) <-> ( u e. A -> u e. Z ) ) )
13	12	biimprd 158	. . . 4 |- ( z = u -> ( ( u e. A -> u e. Z ) -> ( z e. A -> z e. Z ) ) )
14	2, 3, 4, 5, 9, 13	setindis 16102	. . 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		ssalel 3189	. 2 |- ( A C_ Z <-> A. z ( z e. A -> z e. Z ) )
17	15, 16	imbitrrdi 162	1 |- ( A. x e. A ( x = (/) \/ E. y e. A x = suc y ) -> (Ind Z -> A C_ Z ) )

Syntax hints:   -> wi 4   \/ wo 710   A.wal 1371   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   C_ wss 3174   (/)c0 3468   suc csuc 4430  Ind wind 16061

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ral 2491  df-rex 2492  df-v 2778  df-un 3178  df-in 3180  df-ss 3187  df-sn 3649  df-suc 4436  df-bj-ind 16062

This theorem is referenced by:  bj-inf2vn2  16110