Theorem tfindsg2 3153

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 2962	. . 3 |- ((A e. On /\ B e. A) -> B e. On)
2		sucelon 3058	. . 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 2948	. . . 4 |- (A e. On -> Ord A)
5		ordsucss 3059	. . . 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 3061	. . . . . . . . . . 11 |- ((B e. On /\ Ord y) -> (B e. y <-> suc B (_ y))
15		eloni 2948	. . . . . . . . . . 11 |- (y e. On -> Ord y)
16	14, 15	sylan2 451	. . . . . . . . . 10 |- ((B e. On /\ y e. On) -> (B e. y <-> suc B (_ y))
17	16	ancoms 436	. . . . . . . . 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 389	. . . . . . . . 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 453	. . . . . . 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 389	. . . . . . . . 9 |- ((Lim x /\ B e. On) -> (B e. x -> (A.y e. x (B e. y -> ch) -> ph)))
27		ordelsuc 3061	. . . . . . . . . . . . 13 |- ((B e. On /\ Ord x) -> (B e. x <-> suc B (_ x))
28		eloni 2948	. . . . . . . . . . . . 13 |- (x e. On -> Ord x)
29	27, 28	sylan2 451	. . . . . . . . . . . 12 |- ((B e. On /\ x e. On) -> (B e. x <-> suc B (_ x))
30		onelon 2962	. . . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . . 16 |- ((B e. On /\ (x e. On /\ y e. x)) -> (B e. y <-> suc B (_ y))
33	32	anassrs 441	. . . . . . . . . . . . . . 15 |- (((B e. On /\ x e. On) /\ y e. x) -> (B e. y <-> suc B (_ y))
34	33	imbi1d 611	. . . . . . . . . . . . . 14 |- (((B e. On /\ x e. On) /\ y e. x) -> ((B e. y -> ch) <-> (suc B (_ y -> ch)))
35	34	ralbidva 1651	. . . . . . . . . . . . 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 611	. . . . . . . . . . . 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 624	. . . . . . . . . . 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 1804	. . . . . . . . . . . 12 |- x e. V
39		limelon 3022	. . . . . . . . . . . 12 |- ((x e. V /\ Lim x) -> x e. On)
40	38, 39	mpan 693	. . . . . . . . . . 11 |- (Lim x -> x e. On)
41	37, 40	sylan2 451	. . . . . . . . . 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 436	. . . . . . . . 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 453	. . . . . . 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 3152	. . . . 5 |- (((A e. On /\ suc B e. On) /\ suc B (_ A) -> ta)
47	46	exp31 376	. . . 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 389	. 2 |- ((A e. On /\ B e. A) -> ((suc B e. On /\ suc B (_ A) -> ta))
50	3, 7, 49	mp2and 701	1 |- ((A e. On /\ B e. A) -> ta)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 953   e. wcel 955  A.wral 1637  Vcvv 1802   (_ wss 2037  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940

This theorem is referenced by:  oeordi 4198

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-nul 2700  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 774  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-sbc 1932  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-if 2352  df-pw 2392  df-sn 2402  df-pr 2403  df-tp 2405  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-tr 2671  df-eprel 2821  df-po 2831  df-so 2841  df-fr 2907  df-we 2924  df-ord 2941  df-on 2942  df-lim 2943  df-suc 2944

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