Theorem tfis 3938

Description: Transfinite Induction Schema. If all ordinal numbers less than a given number x have a property (induction hypothesis), then all ordinal numbers have the property (conclusion). Exercise 25 of [Enderton] p. 200.

Hypothesis

Ref	Expression
tfis.1	|- (x e. On -> (A.y e. x [y / x]ph -> ph))

Assertion

Ref	Expression
tfis	|- (x e. On -> ph)

Distinct variable groups:   ph,y   x,y

Proof of Theorem tfis

Step	Hyp	Ref	Expression
1		ssrab2 2692	. . . . 5 |- {x e. On | ph} C_ On
2		ax-17 1317	. . . . . . . . . . 11 |- (z e. On -> A.x z e. On)
3		ax-17 1317	. . . . . . . . . . . . 13 |- (y e. z -> A.x y e. z)
4		hbs1 1722	. . . . . . . . . . . . 13 |- ([y / x]ph -> A.x[y / x]ph)
5	3, 4	hbral 2146	. . . . . . . . . . . 12 |- (A.y e. z [y / x]ph -> A.xA.y e. z [y / x]ph)
6		hbs1 1722	. . . . . . . . . . . 12 |- ([z / x]ph -> A.x[z / x]ph)
7	5, 6	hbim 1354	. . . . . . . . . . 11 |- ((A.y e. z [y / x]ph -> [z / x]ph) -> A.x(A.y e. z [y / x]ph -> [z / x]ph))
8	2, 7	hbim 1354	. . . . . . . . . 10 |- ((z e. On -> (A.y e. z [y / x]ph -> [z / x]ph)) -> A.x(z e. On -> (A.y e. z [y / x]ph -> [z / x]ph)))
9		eleq1 1957	. . . . . . . . . . 11 |- (x = z -> (x e. On <-> z e. On))
10		raleq 2266	. . . . . . . . . . . 12 |- (x = z -> (A.y e. x [y / x]ph <-> A.y e. z [y / x]ph))
11		sbequ12 1545	. . . . . . . . . . . 12 |- (x = z -> (ph <-> [z / x]ph))
12	10, 11	imbi12d 688	. . . . . . . . . . 11 |- (x = z -> ((A.y e. x [y / x]ph -> ph) <-> (A.y e. z [y / x]ph -> [z / x]ph)))
13	9, 12	imbi12d 688	. . . . . . . . . 10 |- (x = z -> ((x e. On -> (A.y e. x [y / x]ph -> ph)) <-> (z e. On -> (A.y e. z [y / x]ph -> [z / x]ph))))
14		tfis.1	. . . . . . . . . 10 |- (x e. On -> (A.y e. x [y / x]ph -> ph))
15	8, 13, 14	chvar 1530	. . . . . . . . 9 |- (z e. On -> (A.y e. z [y / x]ph -> [z / x]ph))
16		dfss3 2611	. . . . . . . . . 10 |- (z C_ {x e. On | ph} <-> A.y e. z y e. {x e. On | ph})
17	2	elrabsf 2486	. . . . . . . . . . . 12 |- (y e. {x e. On | ph} <-> (y e. On /\ [y / x]ph))
18	17	simprbi 353	. . . . . . . . . . 11 |- (y e. {x e. On | ph} -> [y / x]ph)
19	18	ralimi 2168	. . . . . . . . . 10 |- (A.y e. z y e. {x e. On | ph} -> A.y e. z [y / x]ph)
20	16, 19	sylbi 216	. . . . . . . . 9 |- (z C_ {x e. On | ph} -> A.y e. z [y / x]ph)
21	15, 20	syl5 20	. . . . . . . 8 |- (z e. On -> (z C_ {x e. On | ph} -> [z / x]ph))
22	21	anc2li 326	. . . . . . 7 |- (z e. On -> (z C_ {x e. On | ph} -> (z e. On /\ [z / x]ph)))
23	2	elrabsf 2486	. . . . . . 7 |- (z e. {x e. On | ph} <-> (z e. On /\ [z / x]ph))
24	22, 23	syl6ibr 230	. . . . . 6 |- (z e. On -> (z C_ {x e. On | ph} -> z e. {x e. On | ph}))
25	24	rgen 2159	. . . . 5 |- A.z e. On (z C_ {x e. On | ph} -> z e. {x e. On | ph})
26		tfi 3937	. . . . 5 |- (({x e. On | ph} C_ On /\ A.z e. On (z C_ {x e. On | ph} -> z e. {x e. On | ph})) -> {x e. On | ph} = On)
27	1, 25, 26	mp2an 761	. . . 4 |- {x e. On | ph} = On
28	27	eqcomi 1888	. . 3 |- On = {x e. On | ph}
29	28	rabeq2i 2291	. 2 |- (x e. On <-> (x e. On /\ ph))
30	29	simprbi 353	1 |- (x e. On -> ph)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  [wsbc 1534  A.wral 2105  {crab 2108   C_ wss 2593  Oncon0 3657

This theorem is referenced by:  tfis2f 3939

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-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

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