Theorem tfis 6662

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 3571	. . . . 5 |- { x e. On | ph } C_ On
2		nfcv 2616	. . . . . . 7 |- F/_ x z
3		nfrab1 3035	. . . . . . . . 9 |- F/_ x { x e. On | ph }
4	2, 3	nfss 3482	. . . . . . . 8 |- F/ x z C_ { x e. On | ph }
5	3	nfcri 2609	. . . . . . . 8 |- F/ x z e. { x e. On | ph }
6	4, 5	nfim 1925	. . . . . . 7 |- F/ x ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
7		dfss3 3479	. . . . . . . . 9 |- ( x C_ { x e. On | ph } <-> A. y e. x y e. { x e. On | ph } )
8		sseq1 3510	. . . . . . . . 9 |- ( x = z -> ( x C_ { x e. On | ph } <-> z C_ { x e. On | ph } ) )
9	7, 8	syl5bbr 259	. . . . . . . 8 |- ( x = z -> ( A. y e. x y e. { x e. On | ph } <-> z C_ { x e. On | ph } ) )
10		rabid 3031	. . . . . . . . 9 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
11		eleq1 2526	. . . . . . . . 9 |- ( x = z -> ( x e. { x e. On | ph } <-> z e. { x e. On | ph } ) )
12	10, 11	syl5bbr 259	. . . . . . . 8 |- ( x = z -> ( ( x e. On /\ ph ) <-> z e. { x e. On | ph } ) )
13	9, 12	imbi12d 318	. . . . . . 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 2119	. . . . . . . . . . . 12 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
15		nfcv 2616	. . . . . . . . . . . . 13 |- F/_ x On
16		nfcv 2616	. . . . . . . . . . . . 13 |- F/_ w On
17		nfv 1712	. . . . . . . . . . . . 13 |- F/ w ph
18		nfs1v 2183	. . . . . . . . . . . . 13 |- F/ x [ w / x ] ph
19		sbequ12 1997	. . . . . . . . . . . . 13 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
20	15, 16, 17, 18, 19	cbvrab 3104	. . . . . . . . . . . 12 |- { x e. On | ph } = { w e. On | [ w / x ] ph }
21	14, 20	elrab2 3256	. . . . . . . . . . 11 |- ( y e. { x e. On | ph } <-> ( y e. On /\ [ y / x ] ph ) )
22	21	simprbi 462	. . . . . . . . . 10 |- ( y e. { x e. On | ph } -> [ y / x ] ph )
23	22	ralimi 2847	. . . . . . . . 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 555	. . . . . . 7 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ( x e. On /\ ph ) ) )
27	2, 6, 13, 26	vtoclgaf 3169	. . . . . 6 |- ( z e. On -> ( z C_ { x e. On | ph } -> z e. { x e. On | ph } ) )
28	27	rgen 2814	. . . . 5 |- A. z e. On ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
29		tfi 6661	. . . . 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 670	. . . 4 |- { x e. On | ph } = On
31	30	eqcomi 2467	. . 3 |- On = { x e. On | ph }
32	31	rabeq2i 3103	. 2 |- ( x e. On <-> ( x e. On /\ ph ) )
33	32	simprbi 462	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   /\ wa 367   = wceq 1398   [wsb 1744   e. wcel 1823   A.wral 2804   {crab 2808   C_ wss 3461   Oncon0 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871

This theorem is referenced by:  tfis2f  6663