Theorem tfindsg2 3169

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 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 2978	. . 3 |- ((A e. On /\ B e. A) -> B e. On)
2		sucelon 3074	. . 3 |- (B e. On <-> suc B e. On)
3	1, 2	sylib 198	. 2 |- ((A e. On /\ B e. A) -> suc B e. On)
4		eloni 2964	. . . 4 |- (A e. On -> Ord A)
5		ordsucss 3075	. . . 4 |- (Ord A -> (B e. A -> suc B (_ A))
6	4, 5	syl 10	. . 3 |- (A e. On -> (B e. A -> suc B (_ A))
7	6	imp 350	. 2 |- ((A e. On /\ B e. A) -> suc B (_ A)
8		tfindsg2.1	. . . . . 6 |- (x = suc B -> (ph <-> ps))
9		tfindsg2.2	. . . . . 6 |- (x = y -> (ph <-> ch))
10		tfindsg2.3	. . . . . 6 |- (x = suc y -> (ph <-> th))
11		tfindsg2.4	. . . . . 6 |- (x = A -> (ph <-> ta))
12		tfindsg2.5	. . . . . . 7 |- (B e. On -> ps)
13	2, 12	sylbir 201	. . . . . 6 |- (suc B e. On -> ps)
14		ordelsuc 3077	. . . . . . . . . . 11 |- ((B e. On /\ Ord y) -> (B e. y <-> suc B (_ y))
15		eloni 2964	. . . . . . . . . . 11 |- (y e. On -> Ord y)
16	14, 15	sylan2 453	. . . . . . . . . 10 |- ((B e. On /\ y e. On) -> (B e. y <-> suc B (_ y))
17	16	ancoms 438	. . . . . . . . 9 |- ((y e. On /\ B e. On) -> (B e. y <-> suc B (_ y))
18		tfindsg2.6	. . . . . . . . . . 11 |- ((y e. On /\ B e. y) -> (ch -> th))
19	18	ex 373	. . . . . . . . . 10 |- (y e. On -> (B e. y -> (ch -> th)))
20	19	adantr 391	. . . . . . . . 9 |- ((y e. On /\ B e. On) -> (B e. y -> (ch -> th)))
21	17, 20	sylbird 205	. . . . . . . 8 |- ((y e. On /\ B e. On) -> (suc B (_ y -> (ch -> th)))
22	21, 2	sylan2br 455	. . . . . . 7 |- ((y e. On /\ suc B e. On) -> (suc B (_ y -> (ch -> th)))
23	22	imp 350	. . . . . 6 |- (((y e. On /\ suc B e. On) /\ suc B (_ y) -> (ch -> th))
24		tfindsg2.7	. . . . . . . . . . 11 |- ((Lim x /\ B e. x) -> (A.y e. x (B e. y -> ch) -> ph))
25	24	ex 373	. . . . . . . . . 10 |- (Lim x -> (B e. x -> (A.y e. x (B e. y -> ch) -> ph)))
26	25	adantr 391	. . . . . . . . 9 |- ((Lim x /\ B e. On) -> (B e. x -> (A.y e. x (B e. y -> ch) -> ph)))
27		ordelsuc 3077	. . . . . . . . . . . . 13 |- ((B e. On /\ Ord x) -> (B e. x <-> suc B (_ x))
28		eloni 2964	. . . . . . . . . . . . 13 |- (x e. On -> Ord x)
29	27, 28	sylan2 453	. . . . . . . . . . . 12 |- ((B e. On /\ x e. On) -> (B e. x <-> suc B (_ x))
30		onelon 2978	. . . . . . . . . . . . . . . . . 18 |- ((x e. On /\ y e. x) -> y e. On)
31	30, 15	syl 10	. . . . . . . . . . . . . . . . 17 |- ((x e. On /\ y e. x) -> Ord y)
32	14, 31	sylan2 453	. . . . . . . . . . . . . . . 16 |- ((B e. On /\ (x e. On /\ y e. x)) -> (B e. y <-> suc B (_ y))
33	32	anassrs 443	. . . . . . . . . . . . . . 15 |- (((B e. On /\ x e. On) /\ y e. x) -> (B e. y <-> suc B (_ y))
34	33	imbi1d 615	. . . . . . . . . . . . . 14 |- (((B e. On /\ x e. On) /\ y e. x) -> ((B e. y -> ch) <-> (suc B (_ y -> ch)))
35	34	ralbidva 1662	. . . . . . . . . . . . 13 |- ((B e. On /\ x e. On) -> (A.y e. x (B e. y -> ch) <-> A.y e. x (suc B (_ y -> ch)))
36	35	imbi1d 615	. . . . . . . . . . . 12 |- ((B e. On /\ x e. On) -> ((A.y e. x (B e. y -> ch) -> ph) <-> (A.y e. x (suc B (_ y -> ch) -> ph)))
37	29, 36	imbi12d 628	. . . . . . . . . . 11 |- ((B e. On /\ x e. On) -> ((B e. x -> (A.y e. x (B e. y -> ch) -> ph)) <-> (suc B (_ x -> (A.y e. x (suc B (_ y -> ch) -> ph))))
38		visset 1816	. . . . . . . . . . . 12 |- x e. V
39		limelon 3038	. . . . . . . . . . . 12 |- ((x e. V /\ Lim x) -> x e. On)
40	38, 39	mpan 697	. . . . . . . . . . 11 |- (Lim x -> x e. On)
41	37, 40	sylan2 453	. . . . . . . . . 10 |- ((B e. On /\ Lim x) -> ((B e. x -> (A.y e. x (B e. y -> ch) -> ph)) <-> (suc B (_ x -> (A.y e. x (suc B (_ y -> ch) -> ph))))
42	41	ancoms 438	. . . . . . . . 9 |- ((Lim x /\ B e. On) -> ((B e. x -> (A.y e. x (B e. y -> ch) -> ph)) <-> (suc B (_ x -> (A.y e. x (suc B (_ y -> ch) -> ph))))
43	26, 42	mpbid 195	. . . . . . . 8 |- ((Lim x /\ B e. On) -> (suc B (_ x -> (A.y e. x (suc B (_ y -> ch) -> ph)))
44	43, 2	sylan2br 455	. . . . . . 7 |- ((Lim x /\ suc B e. On) -> (suc B (_ x -> (A.y e. x (suc B (_ y -> ch) -> ph)))
45	44	imp 350	. . . . . 6 |- (((Lim x /\ suc B e. On) /\ suc B (_ x) -> (A.y e. x (suc B (_ y -> ch) -> ph))
46	8, 9, 10, 11, 13, 23, 45	tfindsg 3168	. . . . 5 |- (((A e. On /\ suc B e. On) /\ suc B (_ A) -> ta)
47	46	exp31 378	. . . 4 |- (A e. On -> (suc B e. On -> (suc B (_ A -> ta)))
48	47	imp3a 361	. . 3 |- (A e. On -> ((suc B e. On /\ suc B (_ A) -> ta))
49	48	adantr 391	. 2 |- ((A e. On /\ B e. A) -> ((suc B e. On /\ suc B (_ A) -> ta))
50	3, 7, 49	mp2and 705	1 |- ((A e. On /\ B e. A) -> ta)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 958   e. wcel 960  A.wral 1648  Vcvv 1814   (_ wss 2050  Ord word 2953  Oncon0 2954  Lim wlim 2955  suc csuc 2956

This theorem is referenced by:  oeordi 4220

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960

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