Theorem bj-inf2vnlem1 11865

Description: Lemma for bj-inf2vn 11869. 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		bi2 128	. . . . 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 666	. . . . . 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 118	. . . . 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 107	. . . . . 6 |- ( ( ( x = (/) -> x e. A ) /\ ( E. y e. A x = suc y -> x e. A ) ) -> ( x = (/) -> x e. A ) )
5		eleq1 2150	. . . . . 6 |- ( x = (/) -> ( x e. A <-> (/) e. A ) )
6	4, 5	mpbidi 149	. . . . 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 1389	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. x ( x = (/) -> (/) e. A ) )
9		exim 1535	. . 3 |- ( A. x ( x = (/) -> (/) e. A ) -> ( E. x x = (/) -> E. x (/) e. A ) )
10		0ex 3966	. . . . . 6 |- (/) e. _V
11	10	isseti 2627	. . . . 5 |- E. x x = (/)
12		pm2.27 39	. . . . 5 |- ( E. x x = (/) -> ( ( E. x x = (/) -> E. x (/) e. A ) -> E. x (/) e. A ) )
13	11, 12	ax-mp 7	. . . 4 |- ( ( E. x x = (/) -> E. x (/) e. A ) -> E. x (/) e. A )
14		bj-ex 11663	. . . 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 112	. . . . . 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 1389	. . . 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 2088	. . . . 5 |- suc z = suc z
21		suceq 4229	. . . . . . 7 |- ( y = z -> suc y = suc z )
22	21	eqeq2d 2099	. . . . . 6 |- ( y = z -> ( suc z = suc y <-> suc z = suc z ) )
23	22	rspcev 2722	. . . . 5 |- ( ( z e. A /\ suc z = suc z ) -> E. y e. A suc z = suc y )
24	20, 23	mpan2 416	. . . 4 |- ( z e. A -> E. y e. A suc z = suc y )
25		vex 2622	. . . . . 6 |- z e. _V
26	25	bj-sucex 11814	. . . . 5 |- suc z e. _V
27		eqeq1 2094	. . . . . . 7 |- ( x = suc z -> ( x = suc y <-> suc z = suc y ) )
28	27	rexbidv 2381	. . . . . 6 |- ( x = suc z -> ( E. y e. A x = suc y <-> E. y e. A suc z = suc y ) )
29		eleq1 2150	. . . . . 6 |- ( x = suc z -> ( x e. A <-> suc z e. A ) )
30	28, 29	imbi12d 232	. . . . 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 2712	. . . 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 2445	. 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 11822	. 2 |- (Ind A <-> ( (/) e. A /\ A. z e. A suc z e. A ) )
35	16, 33, 34	sylanbrc 408	1 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )

Syntax hints:   -> wi 4   /\ wa 102   <-> wb 103   \/ wo 664   A.wal 1287   = wceq 1289   E.wex 1426   e. wcel 1438   A.wral 2359   E.wrex 2360   (/)c0 3286   suc csuc 4192  Ind wind 11821

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-nul 3965  ax-pr 4036  ax-un 4260  ax-bd0 11704  ax-bdor 11707  ax-bdex 11710  ax-bdeq 11711  ax-bdel 11712  ax-bdsb 11713  ax-bdsep 11775

This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 3001  df-un 3003  df-nul 3287  df-sn 3452  df-pr 3453  df-uni 3654  df-suc 4198  df-bdc 11732  df-bj-ind 11822

This theorem is referenced by:  bj-inf2vn  11869  bj-inf2vn2  11870