Theorem tfis 2245

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 group(s):   ph,y   x,y

Proof of Theorem tfis

Step	Hyp	Ref	Expression
1		ssrab 1556	. . . . 5 |- {x e. On | ph} (_ On
2		ax-17 925	. . . . . . . . . . 11 |- (z e. On -> A.x z e. On)
3		ax-17 925	. . . . . . . . . . . . 13 |- (y e. z -> A.x y e. z)
4		hbs1 986	. . . . . . . . . . . . 13 |- ([y / x]ph -> A.x[y / x]ph)
5	3, 4	hbral 1236	. . . . . . . . . . . 12 |- (A.y e. z [y / x]ph -> A.xA.y e. z [y / x]ph)
6		hbs1 986	. . . . . . . . . . . 12 |- ([z / x]ph -> A.x[z / x]ph)
7	5, 6	hbim 702	. . . . . . . . . . 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 702	. . . . . . . . . 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 1149	. . . . . . . . . . 11 |- (x = z -> (x e. On <-> z e. On))
10		raleq 1324	. . . . . . . . . . . 12 |- (x = z -> (A.y e. x [y / x]ph <-> A.y e. z [y / x]ph))
11		sbequ12 865	. . . . . . . . . . . 12 |- (x = z -> (ph <-> [z / x]ph))
12	10, 11	imbi12d 474	. . . . . . . . . . 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 474	. . . . . . . . . 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	chv2 850	. . . . . . . . 9 |- (z e. On -> (A.y e. z [y / x]ph -> [z / x]ph))
16		dfss3 1498	. . . . . . . . . 10 |- (z (_ {x e. On | ph} <-> A.y e. z y e. {x e. On | ph})
17	2	elrabsf 1456	. . . . . . . . . . . 12 |- (y e. {x e. On | ph} <-> (y e. On /\ [y / x]ph))
18	17	pm3.27bd 263	. . . . . . . . . . 11 |- (y e. {x e. On | ph} -> [y / x]ph)
19	18	r19.20si 1254	. . . . . . . . . 10 |- (A.y e. z y e. {x e. On | ph} -> A.y e. z [y / x]ph)
20	16, 19	sylbi 174	. . . . . . . . 9 |- (z (_ {x e. On | ph} -> A.y e. z [y / x]ph)
21	15, 20	syl5 22	. . . . . . . 8 |- (z e. On -> (z (_ {x e. On | ph} -> [z / x]ph))
22	21	anc2li 250	. . . . . . 7 |- (z e. On -> (z (_ {x e. On | ph} -> (z e. On /\ [z / x]ph)))
23	2	elrabsf 1456	. . . . . . 7 |- (z e. {x e. On | ph} <-> (z e. On /\ [z / x]ph))
24	22, 23	syl6ibr 186	. . . . . 6 |- (z e. On -> (z (_ {x e. On | ph} -> z e. {x e. On | ph}))
25	24	rgen 1247	. . . . 5 |- A.z e. On (z (_ {x e. On | ph} -> z e. {x e. On | ph})
26		tfi 2244	. . . . 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 520	. . . 4 |- {x e. On | ph} = On
28	27	cleqcomi 1105	. . 3 |- On = {x e. On | ph}
29	28	cleqrabi 1347	. 2 |- (x e. On <-> (x e. On /\ ph))
30	29	pm3.27bd 263	1 |- (x e. On -> ph)

Syntax hints:   -> wi 2   /\ wa 196   = weq 797   e. wel 803  [wsb 852   = wceq 1091   e. wcel 1092  A.wral 1201  {crab 1204   (_ wss 1487  Oncon0 2199

This theorem is referenced by:  tfis2f 2246

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc 1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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