Theorem tfindsg 3944

Description: Transfinite Induction (inference schema), using implicit substititions. 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 C_ y) -> (ch -> th))
tfindsg.7	|- (((Lim x /\ B e. On) /\ B C_ x) -> (A.y e. x (B C_ y -> ch) -> ph))

Assertion

Ref	Expression
tfindsg	|- (((A e. On /\ B e. On) /\ B C_ 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 2639	. . . . . . 7 |- (x = (/) -> (B C_ x <-> B C_ (/)))
2	1	adantl 424	. . . . . 6 |- ((B = (/) /\ x = (/)) -> (B C_ x <-> B C_ (/)))
3		eqeq2 1893	. . . . . . . 8 |- (B = (/) -> (x = B <-> x = (/)))
4		tfindsg.1	. . . . . . . 8 |- (x = B -> (ph <-> ps))
5	3, 4	syl6bir 232	. . . . . . 7 |- (B = (/) -> (x = (/) -> (ph <-> ps)))
6	5	imp 377	. . . . . 6 |- ((B = (/) /\ x = (/)) -> (ph <-> ps))
7	2, 6	imbi12d 688	. . . . 5 |- ((B = (/) /\ x = (/)) -> ((B C_ x -> ph) <-> (B C_ (/) -> ps)))
8	1	imbi1d 675	. . . . . 6 |- (x = (/) -> ((B C_ x -> ph) <-> (B C_ (/) -> ph)))
9		ss0 2902	. . . . . . . . 9 |- (B C_ (/) -> B = (/))
10	9	con3i 114	. . . . . . . 8 |- (-. B = (/) -> -. B C_ (/))
11	10	pm2.21d 94	. . . . . . 7 |- (-. B = (/) -> (B C_ (/) -> (ph <-> ps)))
12	11	pm5.74d 645	. . . . . 6 |- (-. B = (/) -> ((B C_ (/) -> ph) <-> (B C_ (/) -> ps)))
13	8, 12	sylan9bbr 600	. . . . 5 |- ((-. B = (/) /\ x = (/)) -> ((B C_ x -> ph) <-> (B C_ (/) -> ps)))
14	7, 13	pm2.61ian 534	. . . 4 |- (x = (/) -> ((B C_ x -> ph) <-> (B C_ (/) -> ps)))
15	14	imbi2d 674	. . 3 |- (x = (/) -> ((B e. On -> (B C_ x -> ph)) <-> (B e. On -> (B C_ (/) -> ps))))
16		sseq2 2639	. . . . 5 |- (x = y -> (B C_ x <-> B C_ y))
17		tfindsg.2	. . . . 5 |- (x = y -> (ph <-> ch))
18	16, 17	imbi12d 688	. . . 4 |- (x = y -> ((B C_ x -> ph) <-> (B C_ y -> ch)))
19	18	imbi2d 674	. . 3 |- (x = y -> ((B e. On -> (B C_ x -> ph)) <-> (B e. On -> (B C_ y -> ch))))
20		sseq2 2639	. . . . 5 |- (x = suc y -> (B C_ x <-> B C_ suc y))
21		tfindsg.3	. . . . 5 |- (x = suc y -> (ph <-> th))
22	20, 21	imbi12d 688	. . . 4 |- (x = suc y -> ((B C_ x -> ph) <-> (B C_ suc y -> th)))
23	22	imbi2d 674	. . 3 |- (x = suc y -> ((B e. On -> (B C_ x -> ph)) <-> (B e. On -> (B C_ suc y -> th))))
24		sseq2 2639	. . . . 5 |- (x = A -> (B C_ x <-> B C_ A))
25		tfindsg.4	. . . . 5 |- (x = A -> (ph <-> ta))
26	24, 25	imbi12d 688	. . . 4 |- (x = A -> ((B C_ x -> ph) <-> (B C_ A -> ta)))
27	26	imbi2d 674	. . 3 |- (x = A -> ((B e. On -> (B C_ x -> ph)) <-> (B e. On -> (B C_ A -> ta))))
28		tfindsg.5	. . . 4 |- (B e. On -> ps)
29	28	a1d 15	. . 3 |- (B e. On -> (B C_ (/) -> ps))
30		visset 2295	. . . . . . . . . . . . . 14 |- y e. _V
31	30	sucex 3892	. . . . . . . . . . . . 13 |- suc y e. _V
32	31	eqvinc 2387	. . . . . . . . . . . 12 |- (suc y = B <-> E.x(x = suc y /\ x = B))
33	4, 28	syl5bir 227	. . . . . . . . . . . . . 14 |- (x = B -> (B e. On -> ph))
34	21	biimpd 170	. . . . . . . . . . . . . 14 |- (x = suc y -> (ph -> th))
35	33, 34	sylan9r 519	. . . . . . . . . . . . 13 |- ((x = suc y /\ x = B) -> (B e. On -> th))
36	35	19.23aiv 1674	. . . . . . . . . . . 12 |- (E.x(x = suc y /\ x = B) -> (B e. On -> th))
37	32, 36	sylbi 216	. . . . . . . . . . 11 |- (suc y = B -> (B e. On -> th))
38	37	eqcoms 1887	. . . . . . . . . 10 |- (B = suc y -> (B e. On -> th))
39	38	imim2i 11	. . . . . . . . 9 |- ((B C_ suc y -> B = suc y) -> (B C_ suc y -> (B e. On -> th)))
40	39	a1d 15	. . . . . . . 8 |- ((B C_ suc y -> B = suc y) -> ((B C_ y -> ch) -> (B C_ suc y -> (B e. On -> th))))
41	40	com4r 45	. . . . . . 7 |- (B e. On -> ((B C_ suc y -> B = suc y) -> ((B C_ y -> ch) -> (B C_ suc y -> th))))
42	41	adantl 424	. . . . . 6 |- ((y e. On /\ B e. On) -> ((B C_ suc y -> B = suc y) -> ((B C_ y -> ch) -> (B C_ suc y -> th))))
43		onsssuc 3757	. . . . . . . . . 10 |- ((B e. On /\ y e. On) -> (B C_ y <-> B e. suc y))
44		onelpss 3703	. . . . . . . . . . 11 |- ((B e. On /\ suc y e. On) -> (B e. suc y <-> (B C_ suc y /\ B =/= suc y)))
45		suceloni 3894	. . . . . . . . . . 11 |- (y e. On -> suc y e. On)
46	44, 45	sylan2 500	. . . . . . . . . 10 |- ((B e. On /\ y e. On) -> (B e. suc y <-> (B C_ suc y /\ B =/= suc y)))
47	43, 46	bitrd 587	. . . . . . . . 9 |- ((B e. On /\ y e. On) -> (B C_ y <-> (B C_ suc y /\ B =/= suc y)))
48	47	ancoms 484	. . . . . . . 8 |- ((y e. On /\ B e. On) -> (B C_ y <-> (B C_ suc y /\ B =/= suc y)))
49		tfindsg.6	. . . . . . . . . . . 12 |- (((y e. On /\ B e. On) /\ B C_ y) -> (ch -> th))
50	49	ex 402	. . . . . . . . . . 11 |- ((y e. On /\ B e. On) -> (B C_ y -> (ch -> th)))
51		ax-1 4	. . . . . . . . . . 11 |- (th -> (B C_ suc y -> th))
52	50, 51	syl8 27	. . . . . . . . . 10 |- ((y e. On /\ B e. On) -> (B C_ y -> (ch -> (B C_ suc y -> th))))
53	52	a2d 16	. . . . . . . . 9 |- ((y e. On /\ B e. On) -> ((B C_ y -> ch) -> (B C_ y -> (B C_ suc y -> th))))
54	53	com23 36	. . . . . . . 8 |- ((y e. On /\ B e. On) -> (B C_ y -> ((B C_ y -> ch) -> (B C_ suc y -> th))))
55	48, 54	sylbird 222	. . . . . . 7 |- ((y e. On /\ B e. On) -> ((B C_ suc y /\ B =/= suc y) -> ((B C_ y -> ch) -> (B C_ suc y -> th))))
56		df-ne 2019	. . . . . . . . 9 |- (B =/= suc y <-> -. B = suc y)
57	56	anbi2i 538	. . . . . . . 8 |- ((B C_ suc y /\ B =/= suc y) <-> (B C_ suc y /\ -. B = suc y))
58		annim 257	. . . . . . . 8 |- ((B C_ suc y /\ -. B = suc y) <-> -. (B C_ suc y -> B = suc y))
59	57, 58	bitri 190	. . . . . . 7 |- ((B C_ suc y /\ B =/= suc y) <-> -. (B C_ suc y -> B = suc y))
60	55, 59	syl5ibr 224	. . . . . 6 |- ((y e. On /\ B e. On) -> (-. (B C_ suc y -> B = suc y) -> ((B C_ y -> ch) -> (B C_ suc y -> th))))
61	42, 60	pm2.61d 141	. . . . 5 |- ((y e. On /\ B e. On) -> ((B C_ y -> ch) -> (B C_ suc y -> th)))
62	61	ex 402	. . . 4 |- (y e. On -> (B e. On -> ((B C_ y -> ch) -> (B C_ suc y -> th))))
63	62	a2d 16	. . 3 |- (y e. On -> ((B e. On -> (B C_ y -> ch)) -> (B e. On -> (B C_ suc y -> th))))
64		pm2.27 76	. . . . . . . . 9 |- (B e. On -> ((B e. On -> (B C_ y -> ch)) -> (B C_ y -> ch)))
65	64	ralimdv 2172	. . . . . . . 8 |- (B e. On -> (A.y e. x (B e. On -> (B C_ y -> ch)) -> A.y e. x (B C_ y -> ch)))
66	65	ad2antlr 441	. . . . . . 7 |- (((Lim x /\ B e. On) /\ B C_ x) -> (A.y e. x (B e. On -> (B C_ y -> ch)) -> A.y e. x (B C_ y -> ch)))
67		tfindsg.7	. . . . . . 7 |- (((Lim x /\ B e. On) /\ B C_ x) -> (A.y e. x (B C_ y -> ch) -> ph))
68	66, 67	syld 30	. . . . . 6 |- (((Lim x /\ B e. On) /\ B C_ x) -> (A.y e. x (B e. On -> (B C_ y -> ch)) -> ph))
69	68	exp31 407	. . . . 5 |- (Lim x -> (B e. On -> (B C_ x -> (A.y e. x (B e. On -> (B C_ y -> ch)) -> ph))))
70	69	com3l 38	. . . 4 |- (B e. On -> (B C_ x -> (Lim x -> (A.y e. x (B e. On -> (B C_ y -> ch)) -> ph))))
71	70	com4t 44	. . 3 |- (Lim x -> (A.y e. x (B e. On -> (B C_ y -> ch)) -> (B e. On -> (B C_ x -> ph))))
72	15, 19, 23, 27, 29, 63, 71	tfinds 3942	. 2 |- (A e. On -> (B e. On -> (B C_ A -> ta)))
73	72	imp31 389	1 |- (((A e. On /\ B e. On) /\ B C_ A) -> ta)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  A.wral 2105   C_ wss 2593  (/)c0 2875  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  tfindsg2 3945  oaordi 5227  oaabs 5309  infensuc 5745  r1ord 5766

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-sbc 2454  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663

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