Theorem tfindsg2 3945

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 suc B instead of zero.

Hypotheses

Ref	Expression
tfindsg2.1	|- (x = suc B -> (ph <-> ps))
tfindsg2.2	|- (x = y -> (ph <-> ch))
tfindsg2.3	|- (x = suc y -> (ph <-> th))
tfindsg2.4	|- (x = A -> (ph <-> ta))
tfindsg2.5	|- (B e. On -> ps)
tfindsg2.6	|- ((y e. On /\ B e. y) -> (ch -> th))
tfindsg2.7	|- ((Lim x /\ B e. x) -> (A.y e. x (B e. y -> ch) -> ph))

Assertion

Ref	Expression
tfindsg2	|- ((A e. On /\ B e. A) -> ta)

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

Proof of Theorem tfindsg2

Step	Hyp	Ref	Expression
1		onelon 3683	. . 3 |- ((A e. On /\ B e. A) -> B e. On)
2		sucelon 3898	. . 3 |- (B e. On <-> suc B e. On)
3	1, 2	sylib 215	. 2 |- ((A e. On /\ B e. A) -> suc B e. On)
4		eloni 3667	. . . 4 |- (A e. On -> Ord A)
5		ordsucss 3899	. . . 4 |- (Ord A -> (B e. A -> suc B C_ A))
6	4, 5	syl 12	. . 3 |- (A e. On -> (B e. A -> suc B C_ A))
7	6	imp 377	. 2 |- ((A e. On /\ B e. A) -> suc B C_ A)
8		tfindsg2.1	. . . . 5 |- (x = suc B -> (ph <-> ps))
9		tfindsg2.2	. . . . 5 |- (x = y -> (ph <-> ch))
10		tfindsg2.3	. . . . 5 |- (x = suc y -> (ph <-> th))
11		tfindsg2.4	. . . . 5 |- (x = A -> (ph <-> ta))
12		tfindsg2.5	. . . . . 6 |- (B e. On -> ps)
13	2, 12	sylbir 218	. . . . 5 |- (suc B e. On -> ps)
14		ordelsuc 3900	. . . . . . . . . 10 |- ((B e. On /\ Ord y) -> (B e. y <-> suc B C_ y))
15		eloni 3667	. . . . . . . . . 10 |- (y e. On -> Ord y)
16	14, 15	sylan2 500	. . . . . . . . 9 |- ((B e. On /\ y e. On) -> (B e. y <-> suc B C_ y))
17	16	ancoms 484	. . . . . . . 8 |- ((y e. On /\ B e. On) -> (B e. y <-> suc B C_ y))
18		tfindsg2.6	. . . . . . . . . 10 |- ((y e. On /\ B e. y) -> (ch -> th))
19	18	ex 402	. . . . . . . . 9 |- (y e. On -> (B e. y -> (ch -> th)))
20	19	adantr 425	. . . . . . . 8 |- ((y e. On /\ B e. On) -> (B e. y -> (ch -> th)))
21	17, 20	sylbird 222	. . . . . . 7 |- ((y e. On /\ B e. On) -> (suc B C_ y -> (ch -> th)))
22	21, 2	sylan2br 502	. . . . . 6 |- ((y e. On /\ suc B e. On) -> (suc B C_ y -> (ch -> th)))
23	22	imp 377	. . . . 5 |- (((y e. On /\ suc B e. On) /\ suc B C_ y) -> (ch -> th))
24		tfindsg2.7	. . . . . . . . . 10 |- ((Lim x /\ B e. x) -> (A.y e. x (B e. y -> ch) -> ph))
25	24	ex 402	. . . . . . . . 9 |- (Lim x -> (B e. x -> (A.y e. x (B e. y -> ch) -> ph)))
26	25	adantr 425	. . . . . . . 8 |- ((Lim x /\ B e. On) -> (B e. x -> (A.y e. x (B e. y -> ch) -> ph)))
27		ordelsuc 3900	. . . . . . . . . . . 12 |- ((B e. On /\ Ord x) -> (B e. x <-> suc B C_ x))
28		eloni 3667	. . . . . . . . . . . 12 |- (x e. On -> Ord x)
29	27, 28	sylan2 500	. . . . . . . . . . 11 |- ((B e. On /\ x e. On) -> (B e. x <-> suc B C_ x))
30		onelon 3683	. . . . . . . . . . . . . . . . 17 |- ((x e. On /\ y e. x) -> y e. On)
31	30, 15	syl 12	. . . . . . . . . . . . . . . 16 |- ((x e. On /\ y e. x) -> Ord y)
32	14, 31	sylan2 500	. . . . . . . . . . . . . . 15 |- ((B e. On /\ (x e. On /\ y e. x)) -> (B e. y <-> suc B C_ y))
33	32	anassrs 489	. . . . . . . . . . . . . 14 |- (((B e. On /\ x e. On) /\ y e. x) -> (B e. y <-> suc B C_ y))
34	33	imbi1d 675	. . . . . . . . . . . . 13 |- (((B e. On /\ x e. On) /\ y e. x) -> ((B e. y -> ch) <-> (suc B C_ y -> ch)))
35	34	ralbidva 2119	. . . . . . . . . . . 12 |- ((B e. On /\ x e. On) -> (A.y e. x (B e. y -> ch) <-> A.y e. x (suc B C_ y -> ch)))
36	35	imbi1d 675	. . . . . . . . . . 11 |- ((B e. On /\ x e. On) -> ((A.y e. x (B e. y -> ch) -> ph) <-> (A.y e. x (suc B C_ y -> ch) -> ph)))
37	29, 36	imbi12d 688	. . . . . . . . . 10 |- ((B e. On /\ x e. On) -> ((B e. x -> (A.y e. x (B e. y -> ch) -> ph)) <-> (suc B C_ x -> (A.y e. x (suc B C_ y -> ch) -> ph))))
38		visset 2295	. . . . . . . . . . 11 |- x e. _V
39		limelon 3727	. . . . . . . . . . 11 |- ((x e. _V /\ Lim x) -> x e. On)
40	38, 39	mpan 759	. . . . . . . . . 10 |- (Lim x -> x e. On)
41	37, 40	sylan2 500	. . . . . . . . 9 |- ((B e. On /\ Lim x) -> ((B e. x -> (A.y e. x (B e. y -> ch) -> ph)) <-> (suc B C_ x -> (A.y e. x (suc B C_ y -> ch) -> ph))))
42	41	ancoms 484	. . . . . . . 8 |- ((Lim x /\ B e. On) -> ((B e. x -> (A.y e. x (B e. y -> ch) -> ph)) <-> (suc B C_ x -> (A.y e. x (suc B C_ y -> ch) -> ph))))
43	26, 42	mpbid 212	. . . . . . 7 |- ((Lim x /\ B e. On) -> (suc B C_ x -> (A.y e. x (suc B C_ y -> ch) -> ph)))
44	43, 2	sylan2br 502	. . . . . 6 |- ((Lim x /\ suc B e. On) -> (suc B C_ x -> (A.y e. x (suc B C_ y -> ch) -> ph)))
45	44	imp 377	. . . . 5 |- (((Lim x /\ suc B e. On) /\ suc B C_ x) -> (A.y e. x (suc B C_ y -> ch) -> ph))
46	8, 9, 10, 11, 13, 23, 45	tfindsg 3944	. . . 4 |- (((A e. On /\ suc B e. On) /\ suc B C_ A) -> ta)
47	46	expl 420	. . 3 |- (A e. On -> ((suc B e. On /\ suc B C_ A) -> ta))
48	47	adantr 425	. 2 |- ((A e. On /\ B e. A) -> ((suc B e. On /\ suc B C_ A) -> ta))
49	3, 7, 48	mp2and 767	1 |- ((A e. On /\ B e. A) -> ta)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  _Vcvv 2292   C_ wss 2593  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  oeordi 5262  omsubsdomlem2 5880  omsubsdomlem2OLD 15389

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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