Theorem tfis 7054

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 3687	. . . . 5 |- { x e. On | ph } C_ On
2		nfcv 2764	. . . . . . 7 |- F/_ x z
3		nfrab1 3122	. . . . . . . . 9 |- F/_ x { x e. On | ph }
4	2, 3	nfss 3596	. . . . . . . 8 |- F/ x z C_ { x e. On | ph }
5	3	nfcri 2758	. . . . . . . 8 |- F/ x z e. { x e. On | ph }
6	4, 5	nfim 1825	. . . . . . 7 |- F/ x ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
7		dfss3 3592	. . . . . . . . 9 |- ( x C_ { x e. On | ph } <-> A. y e. x y e. { x e. On | ph } )
8		sseq1 3626	. . . . . . . . 9 |- ( x = z -> ( x C_ { x e. On | ph } <-> z C_ { x e. On | ph } ) )
9	7, 8	syl5bbr 274	. . . . . . . 8 |- ( x = z -> ( A. y e. x y e. { x e. On | ph } <-> z C_ { x e. On | ph } ) )
10		rabid 3116	. . . . . . . . 9 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
11		eleq1 2689	. . . . . . . . 9 |- ( x = z -> ( x e. { x e. On | ph } <-> z e. { x e. On | ph } ) )
12	10, 11	syl5bbr 274	. . . . . . . 8 |- ( x = z -> ( ( x e. On /\ ph ) <-> z e. { x e. On | ph } ) )
13	9, 12	imbi12d 334	. . . . . . 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 2376	. . . . . . . . . . . 12 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
15		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ x On
16		nfcv 2764	. . . . . . . . . . . . 13 |- F/_ w On
17		nfv 1843	. . . . . . . . . . . . 13 |- F/ w ph
18		nfs1v 2437	. . . . . . . . . . . . 13 |- F/ x [ w / x ] ph
19		sbequ12 2111	. . . . . . . . . . . . 13 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
20	15, 16, 17, 18, 19	cbvrab 3198	. . . . . . . . . . . 12 |- { x e. On | ph } = { w e. On | [ w / x ] ph }
21	14, 20	elrab2 3366	. . . . . . . . . . 11 |- ( y e. { x e. On | ph } <-> ( y e. On /\ [ y / x ] ph ) )
22	21	simprbi 480	. . . . . . . . . 10 |- ( y e. { x e. On | ph } -> [ y / x ] ph )
23	22	ralimi 2952	. . . . . . . . 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 34	. . . . . . . 8 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ph ) )
26	25	anc2li 580	. . . . . . 7 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ( x e. On /\ ph ) ) )
27	2, 6, 13, 26	vtoclgaf 3271	. . . . . 6 |- ( z e. On -> ( z C_ { x e. On | ph } -> z e. { x e. On | ph } ) )
28	27	rgen 2922	. . . . 5 |- A. z e. On ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
29		tfi 7053	. . . . 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 708	. . . 4 |- { x e. On | ph } = On
31	30	eqcomi 2631	. . 3 |- On = { x e. On | ph }
32	31	rabeq2i 3197	. 2 |- ( x e. On <-> ( x e. On /\ ph ) )
33	32	simprbi 480	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   [wsb 1880   e. wcel 1990   A.wral 2912   {crab 2916   C_ wss 3574   Oncon0 5723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727

This theorem is referenced by:  tfis2f  7055