Theorem tfi 6638

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 6647 or tfinds 6644 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 3531	. . . . . . . . 9 |- ( x e. ( On \ A ) -> -. x e. A )
2	1	adantl 467	. . . . . . . 8 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. x e. A )
3		eldifi 3530	. . . . . . . . . 10 |- ( x e. ( On \ A ) -> x e. On )
4		onss 6575	. . . . . . . . . . . . 13 |- ( x e. On -> x C_ On )
5		difin0ss 3806	. . . . . . . . . . . . 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 78	. . . . . . . . . . 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 33	. . . . . . . . 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 430	. . . . . . . 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 180	. . . . . . 7 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. ( ( On \ A ) i^i x ) = (/) )
12	11	ex 435	. . . . . 6 |- ( ( x e. On -> ( x C_ A -> x e. A ) ) -> ( x e. ( On \ A ) -> -. ( ( On \ A ) i^i x ) = (/) ) )
13	12	ralimi2 2755	. . . . 5 |- ( A. x e. On ( x C_ A -> x e. A ) -> A. x e. ( On \ A ) -. ( ( On \ A ) i^i x ) = (/) )
14		ralnex 2811	. . . . 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 199	. . . 4 |- ( A. x e. On ( x C_ A -> x e. A ) -> -. E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
16		ssdif0 3796	. . . . . 6 |- ( On C_ A <-> ( On \ A ) = (/) )
17	16	necon3bbii 2648	. . . . 5 |- ( -. On C_ A <-> ( On \ A ) =/= (/) )
18		ordon 6567	. . . . . 6 |- Ord On
19		difss 3535	. . . . . 6 |- ( On \ A ) C_ On
20		tz7.5 5406	. . . . . 6 |- ( ( Ord On /\ ( On \ A ) C_ On /\ ( On \ A ) =/= (/) ) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
21	18, 19, 20	mp3an12 1350	. . . . 5 |- ( ( On \ A ) =/= (/) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
22	17, 21	sylbi 198	. . . 4 |- ( -. On C_ A -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
23	15, 22	nsyl2 130	. . 3 |- ( A. x e. On ( x C_ A -> x e. A ) -> On C_ A )
24	23	anim2i 571	. 2 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> ( A C_ On /\ On C_ A ) )
25		eqss 3422	. 2 |- ( A = On <-> ( A C_ On /\ On C_ A ) )
26	24, 25	sylibr 215	1 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 370   = wceq 1437   e. wcel 1872   =/= wne 2599   A.wral 2714   E.wrex 2715   \ cdif 3376   i^i cin 3378   C_ wss 3379   (/)c0 3704   Ord word 5384   Oncon0 5385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1663  ax-4 1676  ax-5 1752  ax-6 1798  ax-7 1843  ax-8 1874  ax-9 1876  ax-10 1891  ax-11 1896  ax-12 1909  ax-13 2063  ax-ext 2408  ax-sep 4489  ax-nul 4498  ax-pr 4603  ax-un 6541

This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1658  df-nf 1662  df-sb 1791  df-eu 2280  df-mo 2281  df-clab 2415  df-cleq 2421  df-clel 2424  df-nfc 2558  df-ne 2601  df-ral 2719  df-rex 2720  df-rab 2723  df-v 3024  df-sbc 3243  df-dif 3382  df-un 3384  df-in 3386  df-ss 3393  df-pss 3395  df-nul 3705  df-if 3855  df-sn 3942  df-pr 3944  df-tp 3946  df-op 3948  df-uni 4163  df-br 4367  df-opab 4426  df-tr 4462  df-eprel 4707  df-po 4717  df-so 4718  df-fr 4755  df-we 4757  df-ord 5388  df-on 5389

This theorem is referenced by:  tfis  6639  tfisg  30408