Theorem tfindsg 3152

Description: Transfinite Induction (inference schema) with implicit substitutions. The first four hypotheses establish the substitutions we need. The last three are the basis, the induction hypothesis for successors, and the induction hypothesis for limit ordinals. The basis of this version is an arbitrary ordinal B instead of zero. Remark in [TakeutiZaring] p. 57.

Hypotheses

Ref	Expression
tfindsg.1	|- (x = B -> (ph <-> ps))
tfindsg.2	|- (x = y -> (ph <-> ch))
tfindsg.3	|- (x = suc y -> (ph <-> th))
tfindsg.4	|- (x = A -> (ph <-> ta))
tfindsg.5	|- (B e. On -> ps)
tfindsg.6	|- (((y e. On /\ B e. On) /\ B (_ y) -> (ch -> th))
tfindsg.7	|- (((Lim x /\ B e. On) /\ B (_ x) -> (A.y e. x (B (_ y -> ch) -> ph))

Assertion

Ref	Expression
tfindsg	|- (((A e. On /\ B e. On) /\ B (_ A) -> ta)

Distinct variable groups:   x,A   x,y,B   ch,x   th,x   ta,x   ph,y

Proof of Theorem tfindsg

Step	Hyp	Ref	Expression
1		sseq2 2073	. . . . . . 7 |- (x = (/) -> (B (_ x <-> B (_ (/)))
2	1	adantl 388	. . . . . 6 |- ((B = (/) /\ x = (/)) -> (B (_ x <-> B (_ (/)))
3		eqeq2 1476	. . . . . . . 8 |- (B = (/) -> (x = B <-> x = (/)))
4		tfindsg.1	. . . . . . . 8 |- (x = B -> (ph <-> ps))
5	3, 4	syl6bir 215	. . . . . . 7 |- (B = (/) -> (x = (/) -> (ph <-> ps)))
6	5	imp 350	. . . . . 6 |- ((B = (/) /\ x = (/)) -> (ph <-> ps))
7	2, 6	imbi12d 624	. . . . 5 |- ((B = (/) /\ x = (/)) -> ((B (_ x -> ph) <-> (B (_ (/) -> ps)))
8	1	imbi1d 611	. . . . . 6 |- (x = (/) -> ((B (_ x -> ph) <-> (B (_ (/) -> ph)))
9		ss0 2293	. . . . . . . . 9 |- (B (_ (/) -> B = (/))
10	9	con3i 98	. . . . . . . 8 |- (-. B = (/) -> -. B (_ (/))
11	10	pm2.21d 78	. . . . . . 7 |- (-. B = (/) -> (B (_ (/) -> (ph <-> ps)))
12	11	pm5.74d 583	. . . . . 6 |- (-. B = (/) -> ((B (_ (/) -> ph) <-> (B (_ (/) -> ps)))
13	8, 12	sylan9bbr 539	. . . . 5 |- ((-. B = (/) /\ x = (/)) -> ((B (_ x -> ph) <-> (B (_ (/) -> ps)))
14	7, 13	pm2.61ian 475	. . . 4 |- (x = (/) -> ((B (_ x -> ph) <-> (B (_ (/) -> ps)))
15	14	imbi2d 610	. . 3 |- (x = (/) -> ((B e. On -> (B (_ x -> ph)) <-> (B e. On -> (B (_ (/) -> ps))))
16		sseq2 2073	. . . . 5 |- (x = y -> (B (_ x <-> B (_ y))
17		tfindsg.2	. . . . 5 |- (x = y -> (ph <-> ch))
18	16, 17	imbi12d 624	. . . 4 |- (x = y -> ((B (_ x -> ph) <-> (B (_ y -> ch)))
19	18	imbi2d 610	. . 3 |- (x = y -> ((B e. On -> (B (_ x -> ph)) <-> (B e. On -> (B (_ y -> ch))))
20		sseq2 2073	. . . . 5 |- (x = suc y -> (B (_ x <-> B (_ suc y))
21		tfindsg.3	. . . . 5 |- (x = suc y -> (ph <-> th))
22	20, 21	imbi12d 624	. . . 4 |- (x = suc y -> ((B (_ x -> ph) <-> (B (_ suc y -> th)))
23	22	imbi2d 610	. . 3 |- (x = suc y -> ((B e. On -> (B (_ x -> ph)) <-> (B e. On -> (B (_ suc y -> th))))
24		sseq2 2073	. . . . 5 |- (x = A -> (B (_ x <-> B (_ A))
25		tfindsg.4	. . . . 5 |- (x = A -> (ph <-> ta))
26	24, 25	imbi12d 624	. . . 4 |- (x = A -> ((B (_ x -> ph) <-> (B (_ A -> ta)))
27	26	imbi2d 610	. . 3 |- (x = A -> ((B e. On -> (B (_ x -> ph)) <-> (B e. On -> (B (_ A -> ta))))
28		tfindsg.5	. . . 4 |- (B e. On -> ps)
29	28	a1d 12	. . 3 |- (B e. On -> (B (_ (/) -> ps))
30		visset 1804	. . . . . . . . . . . . . 14 |- y e. V
31	30	sucex 3040	. . . . . . . . . . . . 13 |- suc y e. V
32	31	eqvinc 1874	. . . . . . . . . . . 12 |- (suc y = B <-> E.x(x = suc y /\ x = B))
33	4, 28	syl5bir 210	. . . . . . . . . . . . . 14 |- (x = B -> (B e. On -> ph))
34	21	biimpd 153	. . . . . . . . . . . . . 14 |- (x = suc y -> (ph -> th))
35	33, 34	sylan9r 469	. . . . . . . . . . . . 13 |- ((x = suc y /\ x = B) -> (B e. On -> th))
36	35	19.23aiv 1290	. . . . . . . . . . . 12 |- (E.x(x = suc y /\ x = B) -> (B e. On -> th))
37	32, 36	sylbi 199	. . . . . . . . . . 11 |- (suc y = B -> (B e. On -> th))
38	37	eqcoms 1470	. . . . . . . . . 10 |- (B = suc y -> (B e. On -> th))
39	38	imim2i 17	. . . . . . . . 9 |- ((B (_ suc y -> B = suc y) -> (B (_ suc y -> (B e. On -> th)))
40	39	a1d 12	. . . . . . . 8 |- ((B (_ suc y -> B = suc y) -> ((B (_ y -> ch) -> (B (_ suc y -> (B e. On -> th))))
41	40	com4r 41	. . . . . . 7 |- (B e. On -> ((B (_ suc y -> B = suc y) -> ((B (_ y -> ch) -> (B (_ suc y -> th))))
42	41	adantl 388	. . . . . 6 |- ((y e. On /\ B e. On) -> ((B (_ suc y -> B = suc y) -> ((B (_ y -> ch) -> (B (_ suc y -> th))))
43		onsssuc 3048	. . . . . . . . . 10 |- ((B e. On /\ y e. On) -> (B (_ y <-> B e. suc y))
44		onelpsst 2988	. . . . . . . . . . 11 |- ((B e. On /\ suc y e. On) -> (B e. suc y <-> (B (_ suc y /\ B =/= suc y)))
45		suceloni 3052	. . . . . . . . . . 11 |- (y e. On -> suc y e. On)
46	44, 45	sylan2 451	. . . . . . . . . 10 |- ((B e. On /\ y e. On) -> (B e. suc y <-> (B (_ suc y /\ B =/= suc y)))
47	43, 46	bitrd 526	. . . . . . . . 9 |- ((B e. On /\ y e. On) -> (B (_ y <-> (B (_ suc y /\ B =/= suc y)))
48	47	ancoms 436	. . . . . . . 8 |- ((y e. On /\ B e. On) -> (B (_ y <-> (B (_ suc y /\ B =/= suc y)))
49		tfindsg.6	. . . . . . . . . . . 12 |- (((y e. On /\ B e. On) /\ B (_ y) -> (ch -> th))
50	49	ex 373	. . . . . . . . . . 11 |- ((y e. On /\ B e. On) -> (B (_ y -> (ch -> th)))
51		ax-1 4	. . . . . . . . . . 11 |- (th -> (B (_ suc y -> th))
52	50, 51	syl8 24	. . . . . . . . . 10 |- ((y e. On /\ B e. On) -> (B (_ y -> (ch -> (B (_ suc y -> th))))
53	52	a2d 13	. . . . . . . . 9 |- ((y e. On /\ B e. On) -> ((B (_ y -> ch) -> (B (_ y -> (B (_ suc y -> th))))
54	53	com23 32	. . . . . . . 8 |- ((y e. On /\ B e. On) -> (B (_ y -> ((B (_ y -> ch) -> (B (_ suc y -> th))))
55	48, 54	sylbird 205	. . . . . . 7 |- ((y e. On /\ B e. On) -> ((B (_ suc y /\ B =/= suc y) -> ((B (_ y -> ch) -> (B (_ suc y -> th))))
56		df-ne 1579	. . . . . . . . 9 |- (B =/= suc y <-> -. B = suc y)
57	56	anbi2i 479	. . . . . . . 8 |- ((B (_ suc y /\ B =/= suc y) <-> (B (_ suc y /\ -. B = suc y))
58		annim 238	. . . . . . . 8 |- ((B (_ suc y /\ -. B = suc y) <-> -. (B (_ suc y -> B = suc y))
59	57, 58	bitr 173	. . . . . . 7 |- ((B (_ suc y /\ B =/= suc y) <-> -. (B (_ suc y -> B = suc y))
60	55, 59	syl5ibr 207	. . . . . 6 |- ((y e. On /\ B e. On) -> (-. (B (_ suc y -> B = suc y) -> ((B (_ y -> ch) -> (B (_ suc y -> th))))
61	42, 60	pm2.61d 127	. . . . 5 |- ((y e. On /\ B e. On) -> ((B (_ y -> ch) -> (B (_ suc y -> th)))
62	61	ex 373	. . . 4 |- (y e. On -> (B e. On -> ((B (_ y -> ch) -> (B (_ suc y -> th))))
63	62	a2d 13	. . 3 |- (y e. On -> ((B e. On -> (B (_ y -> ch)) -> (B e. On -> (B (_ suc y -> th))))
64		pm2.27 62	. . . . . . . . 9 |- (B e. On -> ((B e. On -> (B (_ y -> ch)) -> (B (_ y -> ch)))
65	64	r19.20sdv 1702	. . . . . . . 8 |- (B e. On -> (A.y e. x (B e. On -> (B (_ y -> ch)) -> A.y e. x (B (_ y -> ch)))
66	65	ad2antlr 405	. . . . . . 7 |- (((Lim x /\ B e. On) /\ B (_ x) -> (A.y e. x (B e. On -> (B (_ y -> ch)) -> A.y e. x (B (_ y -> ch)))
67		tfindsg.7	. . . . . . 7 |- (((Lim x /\ B e. On) /\ B (_ x) -> (A.y e. x (B (_ y -> ch) -> ph))
68	66, 67	syld 27	. . . . . 6 |- (((Lim x /\ B e. On) /\ B (_ x) -> (A.y e. x (B e. On -> (B (_ y -> ch)) -> ph))
69	68	exp31 376	. . . . 5 |- (Lim x -> (B e. On -> (B (_ x -> (A.y e. x (B e. On -> (B (_ y -> ch)) -> ph))))
70	69	com3l 34	. . . 4 |- (B e. On -> (B (_ x -> (Lim x -> (A.y e. x (B e. On -> (B (_ y -> ch)) -> ph))))
71	70	com4t 40	. . 3 |- (Lim x -> (A.y e. x (B e. On ->