Theorem tfi 7053

Description: The Principle of Transfinite Induction. Theorem 7.17 of [TakeutiZaring] p. 39. This principle states that if A is a class of ordinal numbers with the property that every ordinal number included in A also belongs to A, then every ordinal number is in A.

See theorem tfindes 7062 or tfinds 7059 for the version involving basis and induction hypotheses. (Contributed by NM, 18-Feb-2004.)

Assertion

Ref	Expression
tfi	|- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )

Distinct variable group:   x, A

Proof of Theorem tfi

Step	Hyp	Ref	Expression
1		eldifn 3733	. . . . . . . . 9 |- ( x e. ( On \ A ) -> -. x e. A )
2	1	adantl 482	. . . . . . . 8 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. x e. A )
3		eldifi 3732	. . . . . . . . . 10 |- ( x e. ( On \ A ) -> x e. On )
4		onss 6990	. . . . . . . . . . . . 13 |- ( x e. On -> x C_ On )
5		difin0ss 3946	. . . . . . . . . . . . 13 |- ( ( ( On \ A ) i^i x ) = (/) -> ( x C_ On -> x C_ A ) )
6	4, 5	syl5com 31	. . . . . . . . . . . 12 |- ( x e. On -> ( ( ( On \ A ) i^i x ) = (/) -> x C_ A ) )
7	6	imim1d 82	. . . . . . . . . . 11 |- ( x e. On -> ( ( x C_ A -> x e. A ) -> ( ( ( On \ A ) i^i x ) = (/) -> x e. A ) ) )
8	7	a2i 14	. . . . . . . . . 10 |- ( ( x e. On -> ( x C_ A -> x e. A ) ) -> ( x e. On -> ( ( ( On \ A ) i^i x ) = (/) -> x e. A ) ) )
9	3, 8	syl5 34	. . . . . . . . 9 |- ( ( x e. On -> ( x C_ A -> x e. A ) ) -> ( x e. ( On \ A ) -> ( ( ( On \ A ) i^i x ) = (/) -> x e. A ) ) )
10	9	imp 445	. . . . . . . 8 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> ( ( ( On \ A ) i^i x ) = (/) -> x e. A ) )
11	2, 10	mtod 189	. . . . . . 7 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. ( ( On \ A ) i^i x ) = (/) )
12	11	ex 450	. . . . . 6 |- ( ( x e. On -> ( x C_ A -> x e. A ) ) -> ( x e. ( On \ A ) -> -. ( ( On \ A ) i^i x ) = (/) ) )
13	12	ralimi2 2949	. . . . 5 |- ( A. x e. On ( x C_ A -> x e. A ) -> A. x e. ( On \ A ) -. ( ( On \ A ) i^i x ) = (/) )
14		ralnex 2992	. . . . 5 |- ( A. x e. ( On \ A ) -. ( ( On \ A ) i^i x ) = (/) <-> -. E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
15	13, 14	sylib 208	. . . 4 |- ( A. x e. On ( x C_ A -> x e. A ) -> -. E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
16		ssdif0 3942	. . . . . 6 |- ( On C_ A <-> ( On \ A ) = (/) )
17	16	necon3bbii 2841	. . . . 5 |- ( -. On C_ A <-> ( On \ A ) =/= (/) )
18		ordon 6982	. . . . . 6 |- Ord On
19		difss 3737	. . . . . 6 |- ( On \ A ) C_ On
20		tz7.5 5744	. . . . . 6 |- ( ( Ord On /\ ( On \ A ) C_ On /\ ( On \ A ) =/= (/) ) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
21	18, 19, 20	mp3an12 1414	. . . . 5 |- ( ( On \ A ) =/= (/) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
22	17, 21	sylbi 207	. . . 4 |- ( -. On C_ A -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
23	15, 22	nsyl2 142	. . 3 |- ( A. x e. On ( x C_ A -> x e. A ) -> On C_ A )
24	23	anim2i 593	. 2 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> ( A C_ On /\ On C_ A ) )
25		eqss 3618	. 2 |- ( A = On <-> ( A C_ On /\ On C_ A ) )
26	24, 25	sylibr 224	1 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   =/= wne 2794   A.wral 2912   E.wrex 2913   \ cdif 3571   i^i cin 3573   C_ wss 3574   (/)c0 3915   Ord word 5722   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:  tfis  7054  tfisg  31716  onsetrec  42451