Theorem tfi 4792

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 4801 or tfinds 4798 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 3430	. . . . . . . . 9 |- ( x e. ( On \ A ) -> -. x e. A )
2	1	adantl 453	. . . . . . . 8 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. x e. A )
3		eldifi 3429	. . . . . . . . . 10 |- ( x e. ( On \ A ) -> x e. On )
4		onss 4730	. . . . . . . . . . . . 13 |- ( x e. On -> x C_ On )
5		difin0ss 3654	. . . . . . . . . . . . 13 |- ( ( ( On \ A ) i^i x ) = (/) -> ( x C_ On -> x C_ A ) )
6	4, 5	syl5com 28	. . . . . . . . . . . 12 |- ( x e. On -> ( ( ( On \ A ) i^i x ) = (/) -> x C_ A ) )
7	6	imim1d 71	. . . . . . . . . . 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 30	. . . . . . . . 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 419	. . . . . . . 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 170	. . . . . . 7 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. ( ( On \ A ) i^i x ) = (/) )
12	11	ex 424	. . . . . 6 |- ( ( x e. On -> ( x C_ A -> x e. A ) ) -> ( x e. ( On \ A ) -> -. ( ( On \ A ) i^i x ) = (/) ) )
13	12	ralimi2 2738	. . . . 5 |- ( A. x e. On ( x C_ A -> x e. A ) -> A. x e. ( On \ A ) -. ( ( On \ A ) i^i x ) = (/) )
14		ralnex 2676	. . . . 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 189	. . . 4 |- ( A. x e. On ( x C_ A -> x e. A ) -> -. E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
16		ssdif0 3646	. . . . . 6 |- ( On C_ A <-> ( On \ A ) = (/) )
17	16	necon3bbii 2598	. . . . 5 |- ( -. On C_ A <-> ( On \ A ) =/= (/) )
18		ordon 4722	. . . . . 6 |- Ord On
19		difss 3434	. . . . . 6 |- ( On \ A ) C_ On
20		tz7.5 4562	. . . . . 6 |- ( ( Ord On /\ ( On \ A ) C_ On /\ ( On \ A ) =/= (/) ) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
21	18, 19, 20	mp3an12 1269	. . . . 5 |- ( ( On \ A ) =/= (/) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
22	17, 21	sylbi 188	. . . 4 |- ( -. On C_ A -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
23	15, 22	nsyl2 121	. . 3 |- ( A. x e. On ( x C_ A -> x e. A ) -> On C_ A )
24	23	anim2i 553	. 2 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> ( A C_ On /\ On C_ A ) )
25		eqss 3323	. 2 |- ( A = On <-> ( A C_ On /\ On C_ A ) )
26	24, 25	sylibr 204	1 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> A = On )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   = wceq 1649   e. wcel 1721   =/= wne 2567   A.wral 2666   E.wrex 2667   \ cdif 3277   i^i cin 3279   C_ wss 3280   (/)c0 3588   Ord word 4540   Oncon0 4541

This theorem is referenced by:  tfis  4793  tfisg  25418

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