Theorem tfis 4306

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 3025	. . . . 5 |- { x e. On | ph } C_ On
2		nfcv 2178	. . . . . . 7 |- F/_ x z
3		nfrab1 2489	. . . . . . . . 9 |- F/_ x { x e. On | ph }
4	2, 3	nfss 2938	. . . . . . . 8 |- F/ x z C_ { x e. On | ph }
5	3	nfcri 2172	. . . . . . . 8 |- F/ x z e. { x e. On | ph }
6	4, 5	nfim 1464	. . . . . . 7 |- F/ x ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
7		dfss3 2935	. . . . . . . . 9 |- ( x C_ { x e. On | ph } <-> A. y e. x y e. { x e. On | ph } )
8		sseq1 2966	. . . . . . . . 9 |- ( x = z -> ( x C_ { x e. On | ph } <-> z C_ { x e. On | ph } ) )
9	7, 8	syl5bbr 183	. . . . . . . 8 |- ( x = z -> ( A. y e. x y e. { x e. On | ph } <-> z C_ { x e. On | ph } ) )
10		rabid 2485	. . . . . . . . 9 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
11		eleq1 2100	. . . . . . . . 9 |- ( x = z -> ( x e. { x e. On | ph } <-> z e. { x e. On | ph } ) )
12	10, 11	syl5bbr 183	. . . . . . . 8 |- ( x = z -> ( ( x e. On /\ ph ) <-> z e. { x e. On | ph } ) )
13	9, 12	imbi12d 223	. . . . . . 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 1721	. . . . . . . . . . . 12 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
15		nfcv 2178	. . . . . . . . . . . . 13 |- F/_ x On
16		nfcv 2178	. . . . . . . . . . . . 13 |- F/_ w On
17		nfv 1421	. . . . . . . . . . . . 13 |- F/ w ph
18		nfs1v 1815	. . . . . . . . . . . . 13 |- F/ x [ w / x ] ph
19		sbequ12 1654	. . . . . . . . . . . . 13 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
20	15, 16, 17, 18, 19	cbvrab 2555	. . . . . . . . . . . 12 |- { x e. On | ph } = { w e. On | [ w / x ] ph }
21	14, 20	elrab2 2700	. . . . . . . . . . 11 |- ( y e. { x e. On | ph } <-> ( y e. On /\ [ y / x ] ph ) )
22	21	simprbi 260	. . . . . . . . . 10 |- ( y e. { x e. On | ph } -> [ y / x ] ph )
23	22	ralimi 2384	. . . . . . . . 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 28	. . . . . . . 8 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ph ) )
26	25	anc2li 312	. . . . . . 7 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ( x e. On /\ ph ) ) )
27	2, 6, 13, 26	vtoclgaf 2618	. . . . . 6 |- ( z e. On -> ( z C_ { x e. On | ph } -> z e. { x e. On | ph } ) )
28	27	rgen 2374	. . . . 5 |- A. z e. On ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
29		tfi 4305	. . . . 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 402	. . . 4 |- { x e. On | ph } = On
31	30	eqcomi 2044	. . 3 |- On = { x e. On | ph }
32	31	rabeq2i 2554	. 2 |- ( x e. On <-> ( x e. On /\ ph ) )
33	32	simprbi 260	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   /\ wa 97   = wceq 1243   e. wcel 1393   [wsb 1645   A.wral 2306   {crab 2310   C_ wss 2917   Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-in 2924  df-ss 2931  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105

This theorem is referenced by:  tfis2f  4307