Theorem bj-inf2vnlem1 13556

Description: Lemma for bj-inf2vn 13560. 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 129	. . . . 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 700	. . . . . 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 119	. . . . 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 108	. . . . . 6 |- ( ( ( x = (/) -> x e. A ) /\ ( E. y e. A x = suc y -> x e. A ) ) -> ( x = (/) -> x e. A ) )
5		eleq1 2220	. . . . . 6 |- ( x = (/) -> ( x e. A <-> (/) e. A ) )
6	4, 5	mpbidi 150	. . . . 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 1435	. . 3 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> A. x ( x = (/) -> (/) e. A ) )
9		exim 1579	. . 3 |- ( A. x ( x = (/) -> (/) e. A ) -> ( E. x x = (/) -> E. x (/) e. A ) )
10		0ex 4091	. . . . . 6 |- (/) e. _V
11	10	isseti 2720	. . . . 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 13347	. . . 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 113	. . . . . 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 1435	. . . 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 2157	. . . . 5 |- suc z = suc z
21		suceq 4362	. . . . . . 7 |- ( y = z -> suc y = suc z )
22	21	eqeq2d 2169	. . . . . 6 |- ( y = z -> ( suc z = suc y <-> suc z = suc z ) )
23	22	rspcev 2816	. . . . 5 |- ( ( z e. A /\ suc z = suc z ) -> E. y e. A suc z = suc y )
24	20, 23	mpan2 422	. . . 4 |- ( z e. A -> E. y e. A suc z = suc y )
25		vex 2715	. . . . . 6 |- z e. _V
26	25	bj-sucex 13509	. . . . 5 |- suc z e. _V
27		eqeq1 2164	. . . . . . 7 |- ( x = suc z -> ( x = suc y <-> suc z = suc y ) )
28	27	rexbidv 2458	. . . . . 6 |- ( x = suc z -> ( E. y e. A x = suc y <-> E. y e. A suc z = suc y ) )
29		eleq1 2220	. . . . . 6 |- ( x = suc z -> ( x e. A <-> suc z e. A ) )
30	28, 29	imbi12d 233	. . . . 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 2806	. . . 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 2529	. 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 13513	. 2 |- (Ind A <-> ( (/) e. A /\ A. z e. A suc z e. A ) )
35	16, 33, 34	sylanbrc 414	1 |- ( A. x ( x e. A <-> ( x = (/) \/ E. y e. A x = suc y ) ) -> Ind A )

Syntax hints:   -> wi 4   /\ wa 103   <-> wb 104   \/ wo 698   A.wal 1333   = wceq 1335   E.wex 1472   e. wcel 2128   A.wral 2435   E.wrex 2436   (/)c0 3394   suc csuc 4325  Ind wind 13512

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-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-nul 4090  ax-pr 4169  ax-un 4393  ax-bd0 13399  ax-bdor 13402  ax-bdex 13405  ax-bdeq 13406  ax-bdel 13407  ax-bdsb 13408  ax-bdsep 13470

This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-dif 3104  df-un 3106  df-nul 3395  df-sn 3566  df-pr 3567  df-uni 3773  df-suc 4331  df-bdc 13427  df-bj-ind 13513

This theorem is referenced by:  bj-inf2vn  13560  bj-inf2vn2  13561