Theorem tfis 4326

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. (Contributed by NM, 1-Aug-1994.) (Revised by Mario Carneiro, 20-Nov-2016.)

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

Allowed substitution hint:   ph( x)

Proof of Theorem tfis

Dummy variables w z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssrab2 3080	. . . . 5 |- { x e. On | ph } C_ On
2		nfcv 2220	. . . . . . 7 |- F/_ x z
3		nfrab1 2534	. . . . . . . . 9 |- F/_ x { x e. On | ph }
4	2, 3	nfss 2993	. . . . . . . 8 |- F/ x z C_ { x e. On | ph }
5	3	nfcri 2214	. . . . . . . 8 |- F/ x z e. { x e. On | ph }
6	4, 5	nfim 1505	. . . . . . 7 |- F/ x ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
7		dfss3 2990	. . . . . . . . 9 |- ( x C_ { x e. On | ph } <-> A. y e. x y e. { x e. On | ph } )
8		sseq1 3021	. . . . . . . . 9 |- ( x = z -> ( x C_ { x e. On | ph } <-> z C_ { x e. On | ph } ) )
9	7, 8	syl5bbr 192	. . . . . . . 8 |- ( x = z -> ( A. y e. x y e. { x e. On | ph } <-> z C_ { x e. On | ph } ) )
10		rabid 2530	. . . . . . . . 9 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
11		eleq1 2142	. . . . . . . . 9 |- ( x = z -> ( x e. { x e. On | ph } <-> z e. { x e. On | ph } ) )
12	10, 11	syl5bbr 192	. . . . . . . 8 |- ( x = z -> ( ( x e. On /\ ph ) <-> z e. { x e. On | ph } ) )
13	9, 12	imbi12d 232	. . . . . . 7 |- ( x = z -> ( ( A. y e. x y e. { x e. On | ph } -> ( x e. On /\ ph ) ) <-> ( z C_ { x e. On | ph } -> z e. { x e. On | ph } ) ) )
14		sbequ 1762	. . . . . . . . . . . 12 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
15		nfcv 2220	. . . . . . . . . . . . 13 |- F/_ x On
16		nfcv 2220	. . . . . . . . . . . . 13 |- F/_ w On
17		nfv 1462	. . . . . . . . . . . . 13 |- F/ w ph
18		nfs1v 1857	. . . . . . . . . . . . 13 |- F/ x [ w / x ] ph
19		sbequ12 1695	. . . . . . . . . . . . 13 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
20	15, 16, 17, 18, 19	cbvrab 2600	. . . . . . . . . . . 12 |- { x e. On | ph } = { w e. On | [ w / x ] ph }
21	14, 20	elrab2 2752	. . . . . . . . . . 11 |- ( y e. { x e. On | ph } <-> ( y e. On /\ [ y / x ] ph ) )
22	21	simprbi 269	. . . . . . . . . 10 |- ( y e. { x e. On | ph } -> [ y / x ] ph )
23	22	ralimi 2427	. . . . . . . . 9 |- ( A. y e. x y e. { x e. On | ph } -> A. y e. x [ y / x ] ph )
24		tfis.1	. . . . . . . . 9 |- ( x e. On -> ( A. y e. x [ y / x ] ph -> ph ) )
25	23, 24	syl5 32	. . . . . . . 8 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ph ) )
26	25	anc2li 322	. . . . . . 7 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ( x e. On /\ ph ) ) )
27	2, 6, 13, 26	vtoclgaf 2664	. . . . . 6 |- ( z e. On -> ( z C_ { x e. On | ph } -> z e. { x e. On | ph } ) )
28	27	rgen 2417	. . . . 5 |- A. z e. On ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
29		tfi 4325	. . . . 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 )
30	1, 28, 29	mp2an 417	. . . 4 |- { x e. On | ph } = On
31	30	eqcomi 2086	. . 3 |- On = { x e. On | ph }
32	31	rabeq2i 2599	. 2 |- ( x e. On <-> ( x e. On /\ ph ) )
33	32	simprbi 269	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   /\ wa 102   = wceq 1285   e. wcel 1434   [wsb 1686   A.wral 2349   {crab 2353   C_ wss 2974   Oncon0 4120

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-setind 4282

This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-in 2980  df-ss 2987  df-uni 3604  df-tr 3878  df-iord 4123  df-on 4125

This theorem is referenced by:  tfis2f  4327