Theorem bj-inf2vnlem1 15700

Description: Lemma for bj-inf2vn 15704. Remark: unoptimized proof (have to use more deduction style). (Contributed by BJ, 8-Dec-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-inf2vnlem1	|- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )

Distinct variable group:   x, A, y

Proof of Theorem bj-inf2vnlem1

Dummy variable z is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biimpr 130	. . . . 5 |- ( ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> ( ( x = (/) \/ E. y e. A x = suc y ) -> x e. A ) )
2		jaob 711	. . . . . 6 |- ( ( ( x = (/) \/ E. y e. A x = suc y ) -> x e. A ) <-> ( ( x = (/) -> x e. A ) /\ ( E. y e. A x = suc y -> x e. A ) ) )
3	2	biimpi 120	. . . . 5 |- ( ( ( x = (/) \/ E. y e. A x = suc y ) -> x e. A ) -> ( ( x = (/) -> x e. A ) /\ ( E. y e. A x = suc y -> x e. A ) ) )
4		simpl 109	. . . . . 6 |- ( ( ( x = (/) -> x e. A ) /\ ( E. y e. A x = suc y -> x e. A ) ) -> ( x = (/) -> x e. A ) )
5		eleq1 2259	. . . . . 6 |- ( x = (/) -> ( x e. A <-> (/) e. A ) )
6	4, 5	mpbidi 151	. . . . 5 |- ( ( ( x = (/) -> x e. A ) /\ ( E. y e. A x = suc y -> x e. A ) ) -> ( x = (/) -> (/) e. A ) )
7	1, 3, 6	3syl 17	. . . 4 |- ( ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> ( x = (/) -> (/) e. A ) )
8	7	alimi 1469	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. x ( x = (/) -> (/) e. A ) )
9		exim 1613	. . 3 |- ( A. x ( x = (/) -> (/) e. A ) -> ( E. x x = (/) -> E. x (/) e. A ) )
10		0ex 4161	. . . . . 6 |- (/) e. _V
11	10	isseti 2771	. . . . 5 |- E. x x = (/)
12		pm2.27 40	. . . . 5 |- ( E. x x = (/) -> ( ( E. x x = (/) -> E. x (/) e. A ) -> E. x (/) e. A ) )
13	11, 12	ax-mp 5	. . . 4 |- ( ( E. x x = (/) -> E. x (/) e. A ) -> E. x (/) e. A )
14		bj-ex 15492	. . . 4 |- ( E. x (/) e. A -> (/) e. A )
15	13, 14	syl 14	. . 3 |- ( ( E. x x = (/) -> E. x (/) e. A ) -> (/) e. A )
16	8, 9, 15	3syl 17	. 2 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> (/) e. A )
17	3	simprd 114	. . . . . 6 |- ( ( ( x = (/) \/ E. y e. A x = suc y ) -> x e. A ) -> ( E. y e. A x = suc y -> x e. A ) )
18	1, 17	syl 14	. . . . 5 |- ( ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> ( E. y e. A x = suc y -> x e. A ) )
19	18	alimi 1469	. . . 4 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. x ( E. y e. A x = suc y -> x e. A ) )
20		eqid 2196	. . . . 5 |- suc z = suc z
21		suceq 4438	. . . . . . 7 |- ( y = z -> suc y = suc z )
22	21	eqeq2d 2208	. . . . . 6 |- ( y = z -> ( suc z = suc y <-> suc z = suc z ) )
23	22	rspcev 2868	. . . . 5 |- ( ( z e. A /\ suc z = suc z ) -> E. y e. A suc z = suc y )
24	20, 23	mpan2 425	. . . 4 |- ( z e. A -> E. y e. A suc z = suc y )
25		vex 2766	. . . . . 6 |- z e. _V
26	25	bj-sucex 15653	. . . . 5 |- suc z e. _V
27		eqeq1 2203	. . . . . . 7 |- ( x = suc z -> ( x = suc y <-> suc z = suc y ) )
28	27	rexbidv 2498	. . . . . 6 |- ( x = suc z -> ( E. y e. A x = suc y <-> E. y e. A suc z = suc y ) )
29		eleq1 2259	. . . . . 6 |- ( x = suc z -> ( x e. A <-> suc z e. A ) )
30	28, 29	imbi12d 234	. . . . 5 |- ( x = suc z -> ( ( E. y e. A x = suc y -> x e. A ) <-> ( E. y e. A suc z = suc y -> suc z e. A ) ) )
31	26, 30	spcv 2858	. . . 4 |- ( A. x ( E. y e. A x = suc y -> x e. A ) -> ( E. y e. A suc z = suc y -> suc z e. A ) )
32	19, 24, 31	syl2im 38	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> ( z e. A -> suc z e. A ) )
33	32	ralrimiv 2569	. 2 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. z e. A suc z e. A )
34		df-bj-ind 15657	. 2 |- (Ind A <-> ( (/) e. A /\ A. z e. A suc z e. A ) )
35	16, 33, 34	sylanbrc 417	1 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   A.wal 1362   = wceq 1364   E.wex 1506   e. wcel 2167   A.wral 2475   E.wrex 2476   (/)c0 3451   suc csuc 4401  Ind wind 15656

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-nul 4160  ax-pr 4243  ax-un 4469  ax-bd0 15543  ax-bdor 15546  ax-bdex 15549  ax-bdeq 15550  ax-bdel 15551  ax-bdsb 15552  ax-bdsep 15614

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-un 3161  df-nul 3452  df-sn 3629  df-pr 3630  df-uni 3841  df-suc 4407  df-bdc 15571  df-bj-ind 15657

This theorem is referenced by:  bj-inf2vn  15704  bj-inf2vn2  15705