Theorem tfis 4793

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 3388	. . . . 5 |- { x e. On | ph } C_ On
2		nfcv 2540	. . . . . . 7 |- F/_ x z
3		nfrab1 2848	. . . . . . . . 9 |- F/_ x { x e. On | ph }
4	2, 3	nfss 3301	. . . . . . . 8 |- F/ x z C_ { x e. On | ph }
5	3	nfcri 2534	. . . . . . . 8 |- F/ x z e. { x e. On | ph }
6	4, 5	nfim 1828	. . . . . . 7 |- F/ x ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
7		dfss3 3298	. . . . . . . . 9 |- ( x C_ { x e. On | ph } <-> A. y e. x y e. { x e. On | ph } )
8		sseq1 3329	. . . . . . . . 9 |- ( x = z -> ( x C_ { x e. On | ph } <-> z C_ { x e. On | ph } ) )
9	7, 8	syl5bbr 251	. . . . . . . 8 |- ( x = z -> ( A. y e. x y e. { x e. On | ph } <-> z C_ { x e. On | ph } ) )
10		rabid 2844	. . . . . . . . 9 |- ( x e. { x e. On | ph } <-> ( x e. On /\ ph ) )
11		eleq1 2464	. . . . . . . . 9 |- ( x = z -> ( x e. { x e. On | ph } <-> z e. { x e. On | ph } ) )
12	10, 11	syl5bbr 251	. . . . . . . 8 |- ( x = z -> ( ( x e. On /\ ph ) <-> z e. { x e. On | ph } ) )
13	9, 12	imbi12d 312	. . . . . . 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 2109	. . . . . . . . . . . 12 |- ( w = y -> ( [ w / x ] ph <-> [ y / x ] ph ) )
15		nfcv 2540	. . . . . . . . . . . . 13 |- F/_ x On
16		nfcv 2540	. . . . . . . . . . . . 13 |- F/_ w On
17		nfv 1626	. . . . . . . . . . . . 13 |- F/ w ph
18		nfs1v 2155	. . . . . . . . . . . . 13 |- F/ x [ w / x ] ph
19		sbequ12 1940	. . . . . . . . . . . . 13 |- ( x = w -> ( ph <-> [ w / x ] ph ) )
20	15, 16, 17, 18, 19	cbvrab 2914	. . . . . . . . . . . 12 |- { x e. On | ph } = { w e. On | [ w / x ] ph }
21	14, 20	elrab2 3054	. . . . . . . . . . 11 |- ( y e. { x e. On | ph } <-> ( y e. On /\ [ y / x ] ph ) )
22	21	simprbi 451	. . . . . . . . . 10 |- ( y e. { x e. On | ph } -> [ y / x ] ph )
23	22	ralimi 2741	. . . . . . . . 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 30	. . . . . . . 8 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ph ) )
26	25	anc2li 541	. . . . . . 7 |- ( x e. On -> ( A. y e. x y e. { x e. On | ph } -> ( x e. On /\ ph ) ) )
27	2, 6, 13, 26	vtoclgaf 2976	. . . . . 6 |- ( z e. On -> ( z C_ { x e. On | ph } -> z e. { x e. On | ph } ) )
28	27	rgen 2731	. . . . 5 |- A. z e. On ( z C_ { x e. On | ph } -> z e. { x e. On | ph } )
29		tfi 4792	. . . . 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 654	. . . 4 |- { x e. On | ph } = On
31	30	eqcomi 2408	. . 3 |- On = { x e. On | ph }
32	31	rabeq2i 2913	. 2 |- ( x e. On <-> ( x e. On /\ ph ) )
33	32	simprbi 451	1 |- ( x e. On -> ph )

Syntax hints:   -> wi 4   /\ wa 359   = wceq 1649   [wsb 1655   e. wcel 1721   A.wral 2666   {crab 2670   C_ wss 3280   Oncon0 4541

This theorem is referenced by:  tfis2f  4794

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-sbc 3122  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-pss 3296  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-tp 3782  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-tr 4263  df-eprel 4454  df-po 4463  df-so 4464  df-fr 4501  df-we 4503  df-ord 4544  df-on 4545