Theorem tfis 3127

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 2131	. . . . 5 |- {x e. On | ph} (_ On
2		ax-17 971	. . . . . . . . . . 11 |- (z e. On -> A.x z e. On)
3		ax-17 971	. . . . . . . . . . . . 13 |- (y e. z -> A.x y e. z)
4		hbs1 1332	. . . . . . . . . . . . 13 |- ([y / x]ph -> A.x[y / x]ph)
5	3, 4	hbral 1686	. . . . . . . . . . . 12 |- (A.y e. z [y / x]ph -> A.xA.y e. z [y / x]ph)
6		hbs1 1332	. . . . . . . . . . . 12 |- ([z / x]ph -> A.x[z / x]ph)
7	5, 6	hbim 1007	. . . . . . . . . . 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 1007	. . . . . . . . . 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 1534	. . . . . . . . . . 11 |- (x = z -> (x e. On <-> z e. On))
10		raleq1 1786	. . . . . . . . . . . 12 |- (x = z -> (A.y e. x [y / x]ph <-> A.y e. z [y / x]ph))
11		sbequ12 1181	. . . . . . . . . . . 12 |- (x = z -> (ph <-> [z / x]ph))
12	10, 11	imbi12d 626	. . . . . . . . . . 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 626	. . . . . . . . . 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 1167	. . . . . . . . 9 |- (z e. On -> (A.y e. z [y / x]ph -> [z / x]ph))
16		dfss3 2059	. . . . . . . . . 10 |- (z (_ {x e. On | ph} <-> A.y e. z y e. {x e. On | ph})
17	2	elrabsf 1963	. . . . . . . . . . . 12 |- (y e. {x e. On | ph} <-> (y e. On /\ [y / x]ph))
18	17	pm3.27bi 326	. . . . . . . . . . 11 |- (y e. {x e. On | ph} -> [y / x]ph)
19	18	r19.20si 1706	. . . . . . . . . 10 |- (A.y e. z y e. {x e. On | ph} -> A.y e. z [y / x]ph)
20	16, 19	sylbi 199	. . . . . . . . 9 |- (z (_ {x e. On | ph} -> A.y e. z [y / x]ph)
21	15, 20	syl5 21	. . . . . . . 8 |- (z e. On -> (z (_ {x e. On | ph} -> [z / x]ph))
22	21	anc2li 302	. . . . . . 7 |- (z e. On -> (z (_ {x e. On | ph} -> (z e. On /\ [z / x]ph)))
23	2	elrabsf 1963	. . . . . . 7 |- (z e. {x e. On | ph} <-> (z e. On /\ [z / x]ph))
24	22, 23	syl6ibr 213	. . . . . 6 |- (z e. On -> (z (_ {x e. On | ph} -> z e. {x e. On | ph}))
25	24	rgen 1698	. . . . 5 |- A.z e. On (z (_ {x e. On | ph} -> z e. {x e. On | ph})
26		tfi 3126	. . . . 5 |- (({x e. On | ph} (_ On /\ A.z e. On (z (_ {x e. On | ph} -> z e. {x e. On | ph})) -> {x e. On | ph} = On)
27	1, 25, 26	mp2an 697	. . . 4 |- {x e. On | ph} = On
28	27	eqcomi 1479	. . 3 |- On = {x e. On | ph}
29	28	rabeq2i 1810	. 2 |- (x e. On <-> (x e. On /\ ph))
30	29	pm3.27bi 326	1 |- (x e. On -> ph)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  [wsbc 1170  A.wral 1645  {crab 1648   (_ wss 2047  Oncon0 2948

This theorem is referenced by:  tfis2f 3128

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-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-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

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