Theorem tfinds 3151

Description: Principle of 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. Theorem Schema 4 of [Suppes] p. 197.

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 2948	. . . . 5 |- (x e. On -> Ord x)
4		df-lim 2943	. . . . . . . . . . . . . . . 16 |- (Lim x <-> (Ord x /\ x =/= (/) /\ x = U.x))
5	4	biimpr 152	. . . . . . . . . . . . . . 15 |- ((Ord x /\ x =/= (/) /\ x = U.x) -> Lim x)
6	5	3com23 837	. . . . . . . . . . . . . 14 |- ((Ord x /\ x = U.x /\ x =/= (/)) -> Lim x)
7	6	3expia 833	. . . . . . . . . . . . 13 |- ((Ord x /\ x = U.x) -> (x =/= (/) -> Lim x))
8	7	necon1bd 1624	. . . . . . . . . . . 12 |- ((Ord x /\ x = U.x) -> (-. Lim x -> x = (/)))
9	8	ex 373	. . . . . . . . . . 11 |- (Ord x -> (x = U.x -> (-. Lim x -> x = (/))))
10	9	com23 32	. . . . . . . . . 10 |- (Ord x -> (-. Lim x -> (x = U.x -> x = (/))))
11		orduninsuc 3104	. . . . . . . . . . 11 |- (Ord x -> (x = U.x <-> -. E.y e. On x = suc y))
12	11	biimprd 154	. . . . . . . . . 10 |- (Ord x -> (-. E.y e. On x = suc y -> x = U.x))
13	10, 12	syl5d 55	. . . . . . . . 9 |- (Ord x -> (-. Lim x -> (-. E.y e. On x = suc y -> x = (/))))
14	13	imp 350	. . . . . . . 8 |- ((Ord x /\ -. Lim x) -> (-. E.y e. On x = suc y -> x = (/)))
15	14	con1d 93	. . . . . . 7 |- ((Ord x /\ -. Lim x) -> (-. x = (/) -> E.y e. On x = suc y))
16	15	orrd 233	. . . . . 6 |- ((Ord x /\ -. Lim x) -> (x = (/) \/ E.y e. On x = suc y))
17	16	ex 373	. . . . 5 |- (Ord x -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
18	3, 17	syl 10	. . . 4 |- (x e. On -> (-. Lim x -> (x = (/) \/ E.y e. On x = suc y)))
19		tfinds.5	. . . . . . 7 |- ps
20		tfinds.1	. . . . . . 7 |- (x = (/) -> (ph <-> ps))
21	19, 20	mpbiri 194	. . . . . 6 |- (x = (/) -> ph)
22	21	a1d 12	. . . . 5 |- (x = (/) -> (A.y e. x ch -> ph))
23		hbra1 1679	. . . . . . 7 |- (A.y e. x ch -> A.yA.y e. x ch)
24		ax-17 968	. . . . . . 7 |- (ph -> A.yph)
25	23, 24	hbim 1004	. . . . . 6 |- ((A.y e. x ch -> ph) -> A.y(A.y e. x ch -> ph))
26		raleq1 1778	. . . . . . . . . . 11 |- (x = suc y -> (A.z e. x [z / x]ph <-> A.z e. suc y[z / x]ph))
27		sbequ 1224	. . . . . . . . . . . . 13 |- (y = z -> ([y / x]ph <-> [z / x]ph))
28		ax-17 968	. . . . . . . . . . . . . 14 |- (ch -> A.xch)
29	28, 1	sbie 1192	. . . . . . . . . . . . 13 |- ([y / x]ph <-> ch)
30	27, 29	syl5bbr 532	. . . . . . . . . . . 12 |- (y = z -> (ch <-> [z / x]ph))
31	30	cbvralv 1791	. . . . . . . . . . 11 |- (A.y e. x ch <-> A.z e. x [z / x]ph)
32		ax-17 968	. . . . . . . . . . . 12 |- (ph -> A.zph)
33		hbs1 1327	. . . . . . . . . . . 12 |- ([z / x]ph -> A.x[z / x]ph)
34		sbequ12 1177	. . . . . . . . . . . 12 |- (x = z -> (ph <-> [z / x]ph))
35	32, 33, 34	cbvral 1789	. . . . . . . . . . 11 |- (A.x e. suc yph <-> A.z e. suc y[z / x]ph)
36	26, 31, 35	3bitr4g 553	. . . . . . . . . 10 |- (x = suc y -> (A.y e. x ch <-> A.x e. suc yph))
37	36	biimpd 153	. . . . . . . . 9 |- (x = suc y -> (A.y e. x ch -> A.x e. suc yph))
38		tfinds.6	. . . . . . . . . 10 |- (y e. On -> (ch -> th))
39		visset 1804	. . . . . . . . . . . 12 |- y e. V
40	39	sucid 3041	. . . . . . . . . . 11 |- y e. suc y
41	1	rcla4v 1864	. . . . . . . . . . 11 |- (y e. suc y -> (A.x e. suc yph -> ch))
42	40, 41	ax-mp 7	. . . . . . . . . 10 |- (A.x e. suc yph -> ch)
43	38, 42	syl5 21	. . . . . . . . 9 |- (y e. On -> (A.x e. suc yph -> th))
44	37, 43	sylan9r 469	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> th))
45		tfinds.3	. . . . . . . . 9 |- (x = suc y -> (ph <-> th))
46	45	adantl 388	. . . . . . . 8 |- ((y e. On /\ x = suc y) -> (ph <-> th))
47	44, 46	sylibrd 204	. . . . . . 7 |- ((y e. On /\ x = suc y) -> (A.y e. x ch -> ph))
48	47	ex 373	. . . . . 6 |- (y e. On -> (x = suc y -> (A.y e. x ch -> ph)))
49	25, 48	r19.23ai 1734	. . . . 5 |- (E.y e. On x = suc y -> (A.y e. x ch -> ph))
50	22, 49	jaoi 341	. . . 4 |- ((x = (/) \/ E.y e. On x = suc y) -> (A.y e. x ch -> ph))
51	18, 50	syl6 22	. . 3 |- (x e. On -> (-. Lim x -> (A.y e. x ch -> ph)))
52		tfinds.7	. . 3 |- (Lim x -> (A.y e. x ch -> ph))
53	51, 52	pm2.61d2 129	. 2 |- (x e. On -> (A.y e. x ch -> ph))
54	1, 2, 53	tfis3 3120	1 |- (A e. On -> ta)

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   \/ wo 222   /\ wa 223   /\ w3a 773   = wceq 953   e. wcel 955  [wsbc 1166   =/= wne 1577  A.wral 1637  E.wrex 1638  (/)c0 2270  U.cuni 2493  Ord word 2937  Oncon0 2938  Lim wlim 2939  suc csuc 2940

This theorem is referenced by:  tfindsg 3152  tfindes 3154  tfinds3 3156  oa0r 4157  om0r 4158  om1r 4161  oe1m 4163  r1tr 4626  alephon 4837  alephcard 4839  alephordi 4846

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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