Theorem tfi 6575

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 6584 or tfinds 6581 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 3588	. . . . . . . . 9 |- ( x e. ( On \ A ) -> -. x e. A )
2	1	adantl 466	. . . . . . . 8 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. x e. A )
3		eldifi 3587	. . . . . . . . . 10 |- ( x e. ( On \ A ) -> x e. On )
4		onss 6513	. . . . . . . . . . . . 13 |- ( x e. On -> x C_ On )
5		difin0ss 3854	. . . . . . . . . . . . 13 |- ( ( ( On \ A ) i^i x ) = (/) -> ( x C_ On -> x C_ A ) )
6	4, 5	syl5com 30	. . . . . . . . . . . 12 |- ( x e. On -> ( ( ( On \ A ) i^i x ) = (/) -> x C_ A ) )
7	6	imim1d 75	. . . . . . . . . . 11 |- ( x e. On -> ( ( x C_ A -> x e. A ) -> ( ( ( On \ A ) i^i x ) = (/) -> x e. A ) ) )
8	7	a2i 13	. . . . . . . . . 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 32	. . . . . . . . 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 429	. . . . . . . 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 177	. . . . . . 7 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. ( ( On \ A ) i^i x ) = (/) )
12	11	ex 434	. . . . . 6 |- ( ( x e. On -> ( x C_ A -> x e. A ) ) -> ( x e. ( On \ A ) -> -. ( ( On \ A ) i^i x ) = (/) ) )
13	12	ralimi2 2816	. . . . 5 |- ( A. x e. On ( x C_ A -> x e. A ) -> A. x e. ( On \ A ) -. ( ( On \ A ) i^i x ) = (/) )
14		ralnex 2852	. . . . 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 196	. . . 4 |- ( A. x e. On ( x C_ A -> x e. A ) -> -. E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
16		ssdif0 3846	. . . . . 6 |- ( On C_ A <-> ( On \ A ) = (/) )
17	16	necon3bbii 2713	. . . . 5 |- ( -. On C_ A <-> ( On \ A ) =/= (/) )
18		ordon 6505	. . . . . 6 |- Ord On
19		difss 3592	. . . . . 6 |- ( On \ A ) C_ On
20		tz7.5 4849	. . . . . 6 |- ( ( Ord On /\ ( On \ A ) C_ On /\ ( On \ A ) =/= (/) ) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
21	18, 19, 20	mp3an12 1305	. . . . 5 |- ( ( On \ A ) =/= (/) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
22	17, 21	sylbi 195	. . . 4 |- ( -. On C_ A -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
23	15, 22	nsyl2 127	. . 3 |- ( A. x e. On ( x C_ A -> x e. A ) -> On C_ A )
24	23	anim2i 569	. 2 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> ( A C_ On /\ On C_ A ) )
25		eqss 3480	. 2 |- ( A = On <-> ( A C_ On /\ On C_ A ) )
26	24, 25	sylibr 212	1 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 369   = wceq 1370   e. wcel 1758   =/= wne 2648   A.wral 2799   E.wrex 2800   \ cdif 3434   i^i cin 3436   C_ wss 3437   (/)c0 3746   Ord word 4827   Oncon0 4828

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-tr 4495  df-eprel 4741  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832

This theorem is referenced by:  tfis  6576  tfisg  27810