Theorem tfindsg2 3163

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 2972	. . 3 |- ((A e. On /\ B e. A) -> B e. On)
2		sucelon 3068	. . 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 2958	. . . 4 |- (A e. On -> Ord A)
5		ordsucss 3069	. . . 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 3071	. . . . . . . . . . 11 |- ((B e. On /\ Ord y) -> (B e. y <-> suc B (_ y))
15		eloni 2958	. . . . . . . . . . 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 3071	. . . . . . . . . . . . 13 |- ((B e. On /\ Ord x) -> (B e. x <-> suc B (_ x))
28		eloni 2958	. . . . . . . . . . . . 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 2972	. . . . . . . . . . . . . . . . . 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 613	. . . . . . . . . . . . . 14 |- (((B e. On /\ x e. On) /\ y e. x) -> ((B e. y -> ch) <-> (suc B (_ y -> ch)))
35	34	ralbidva 1659	. . . . . . . . . . . . 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 613	. . . . . . . . . . . 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 626	. . . . . . . . . . 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 1813	. . . . . . . . . . . 12 |- x e. V
39		limelon 3032	. . . . . . . . . . . 12 |- ((x e. V /\ Lim x) -> x e. On)
40	38, 39	mpan 695	. . . . . . . . . . 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 3162	. . . . 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 703	1 |- ((A e. On /\ B e. A) -> ta)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811   (_ wss 2047  Ord word 2947  Oncon0 2948  Lim wlim 2949  suc csuc 2950

This theorem is referenced by:  oeordi 4214

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954

Copyright terms: Public domain