Theorem tfinds 3942

Description: Principle of Transfinite Induction (inference schema), using 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. Theorem Schema 4 of [Suppes] p. 197. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Hypotheses

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

Assertion

Ref	Expression
tfinds	|- (A e. On -> ta)

Distinct variable groups:   x,y   x,A   ch,x   ta,x   ph,y

Proof of Theorem tfinds

Step	Hyp	Ref	Expression
1		tfinds.2	. 2 |- (x = y -> (ph <-> ch))
2		tfinds.4	. 2 |- (x = A -> (ph <-> ta))
3		eloni 3667	. . . . . . 7 |- (x e. On -> Ord x)
4		pm2.27 76	. . . . . . 7 |- (Ord x -> ((Ord x -> (x = (/) \/ E.y e. On x = suc y)) -> (x = (/) \/ E.y e. On x = suc y)))
5	3, 4	syl 12	. . . . . 6 |- (x e. On -> ((Ord x -> (x = (/) \/ E.y e. On x = suc y)) -> (x = (/) \/ E.y e. On x = suc y)))
6		tfinds.5	. . . . . . . . 9 |- ps
7		tfinds.1	. . . . . . . . 9 |- (x = (/) -> (ph <-> ps))
8	6, 7	mpbiri 211	. . . . . . . 8 |- (x = (/) -> ph)
9	8	a1d 15	. . . . . . 7 |- (x = (/) -> (A.y e. x ch -> ph))
10		hbra1 2147	. . . . . . . . 9 |- (A.y e. x ch -> A.yA.y e. x ch)
11		ax-17 1317	. . . . . . . . 9 |- (ph -> A.yph)
12	10, 11	hbim 1354	. . . . . . . 8 |- ((A.y e. x ch -> ph) -> A.y(A.y e. x ch -> ph))
13		raleq 2266	. . . . . . . . . . . 12 |- (x = suc y -> (A.z e. x [z / x]ph <-> A.z e. suc y[z / x]ph))
14		ax-17 1317	. . . . . . . . . . . . 13 |- (ch -> A.zch)
15		ax-17 1317	. . . . . . . . . . . . 13 |- ([z / x]ph -> A.y[z / x]ph)
16		sbequ 1599	. . . . . . . . . . . . . 14 |- (y = z -> ([y / x]ph <-> [z / x]ph))
17		ax-17 1317	. . . . . . . . . . . . . . 15 |- (ch -> A.xch)
18	17, 1	sbie 1565	. . . . . . . . . . . . . 14 |- ([y / x]ph <-> ch)
19	16, 18	syl5bbr 593	. . . . . . . . . . . . 13 |- (y = z -> (ch <-> [z / x]ph))
20	14, 15, 19	cbvral 2278	. . . . . . . . . . . 12 |- (A.y e. x ch <-> A.z e. x [z / x]ph)
21		ax-17 1317	. . . . . . . . . . . . 13 |- (ph -> A.zph)
22		hbs1 1722	. . . . . . . . . . . . 13 |- ([z / x]ph -> A.x[z / x]ph)
23		sbequ12 1545	. . . . . . . . . . . . 13 |- (x = z -> (ph <-> [z / x]ph))
24	21, 22, 23	cbvral 2278	. . . . . . . . . . . 12 |- (A.x e. suc yph <-> A.z e. suc y[z / x]ph)
25	13, 20, 24	3bitr4g 614	. . . . . . . . . . 11 |- (x = suc y -> (A.y e. x ch <-> A.x e. suc yph))
26	25	imbi1d 675	. . . . . . . . . 10 |- (x = suc y -> ((A.y e. x ch -> th) <-> (A.x e. suc yph -> th)))
27		tfinds.6	. . . . . . . . . . 11 |- (y e. On -> (ch -> th))
28		visset 2295	. . . . . . . . . . . . 13 |- y e. _V
29	28	sucid 3744	. . . . . . . . . . . 12 |- y e. suc y
30	1	rcla4v 2376	. . . . . . . . . . . 12 |- (y e. suc y -> (A.x e. suc yph -> ch))
31	29, 30	ax-mp 7	. . . . . . . . . . 11 |- (A.x e. suc yph -> ch)
32	27, 31	syl5 20	. . . . . . . . . 10 |- (y e. On -> (A.x e. suc yph -> th))
33	26, 32	syl5cbir 228	. . . . . . . . 9 |- (y e. On -> (x = suc y -> (A.y e. x ch -> th)))
34		tfinds.3	. . . . . . . . . . 11 |- (x = suc y -> (ph <-> th))
35	34	biimprd 171	. . . . . . . . . 10 |- (x = suc y -> (th -> ph))
36	35	a1i 8	. . . . . . . . 9 |- (y e. On -> (x = suc y -> (th -> ph)))
37	33, 36	syldd 61	. . . . . . . 8 |- (y e. On -> (x = suc y -> (A.y e. x ch -> ph)))
38	12, 37	r19.23ai 2209	. . . . . . 7 |- (E.y e. On x = suc y -> (A.y e. x ch -> ph))
39	9, 38	jaoi 368	. . . . . 6 |- ((x = (/) \/ E.y e. On x = suc y) -> (A.y e. x ch -> ph))
40	5, 39	syl6 25	. . . . 5 |- (x e. On -> ((Ord x -> (x = (/) \/ E.y e. On x = suc y)) -> (A.y e. x ch -> ph)))
41		iman 256	. . . . 5 |- ((Ord x -> (x = (/) \/ E.y e. On x = suc y)) <-> -. (Ord x /\ -. (x = (/) \/ E.y e. On x = suc y)))
42	40, 41	syl5ibr 224	. . . 4 |- (x e. On -> (-. (Ord x /\ -. (x = (/) \/ E.y e. On x = suc y)) -> (A.y e. x ch -> ph)))
43		dflim3 3930	. . . . 5 |- (Lim x <-> (Ord x /\ -. (x = (/) \/ E.y e. On x = suc y)))
44	43	notbii 204	. . . 4 |- (-. Lim x <-> -. (Ord x /\ -. (x = (/) \/ E.y e. On x = suc y)))
45	42, 44	syl5ib 223	. . 3 |- (x e. On -> (-. Lim x -> (A.y e. x ch -> ph)))
46		tfinds.7	. . 3 |- (Lim x -> (A.y e. x ch -> ph))
47	45, 46	pm2.61d2 143	. 2 |- (x e. On -> (A.y e. x ch -> ph))
48	1, 2, 47	tfis3 3941	1 |- (A e. On -> ta)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   \/ wo 239   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  E.wrex 2106  (/)c0 2875  Ord word 3656  Oncon0 3657  Lim wlim 3658  suc csuc 3659

This theorem is referenced by:  tfindsg 3944  tfindes 3946  tfinds3 3948  oa0r 5220  om0r 5221  om1r 5224  oe1m 5226  oeoalem 5271  r1tr 5765  alephon 5876  elomsubsd 5885  omsublim 5887  alephcard 6015  alephordi 6022  tartarmap 15265  elomsubsdOLD 15394  omsublimOLD 15396  smoge 16454

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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