Theorem tfi 6661

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 6670 or tfinds 6667 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 3613	. . . . . . . . 9 |- ( x e. ( On \ A ) -> -. x e. A )
2	1	adantl 464	. . . . . . . 8 |- ( ( ( x e. On -> ( x C_ A -> x e. A ) ) /\ x e. ( On \ A ) ) -> -. x e. A )
3		eldifi 3612	. . . . . . . . . 10 |- ( x e. ( On \ A ) -> x e. On )
4		onss 6599	. . . . . . . . . . . . 13 |- ( x e. On -> x C_ On )
5		difin0ss 3882	. . . . . . . . . . . . 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 427	. . . . . . . 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 432	. . . . . 6 |- ( ( x e. On -> ( x C_ A -> x e. A ) ) -> ( x e. ( On \ A ) -> -. ( ( On \ A ) i^i x ) = (/) ) )
13	12	ralimi2 2844	. . . . 5 |- ( A. x e. On ( x C_ A -> x e. A ) -> A. x e. ( On \ A ) -. ( ( On \ A ) i^i x ) = (/) )
14		ralnex 2900	. . . . 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 3873	. . . . . 6 |- ( On C_ A <-> ( On \ A ) = (/) )
17	16	necon3bbii 2715	. . . . 5 |- ( -. On C_ A <-> ( On \ A ) =/= (/) )
18		ordon 6591	. . . . . 6 |- Ord On
19		difss 3617	. . . . . 6 |- ( On \ A ) C_ On
20		tz7.5 4888	. . . . . 6 |- ( ( Ord On /\ ( On \ A ) C_ On /\ ( On \ A ) =/= (/) ) -> E. x e. ( On \ A ) ( ( On \ A ) i^i x ) = (/) )
21	18, 19, 20	mp3an12 1312	. . . . 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 567	. 2 |- ( ( A C_ On /\ A. x e. On ( x C_ A -> x e. A ) ) -> ( A C_ On /\ On C_ A ) )
25		eqss 3504	. 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 367   = wceq 1398   e. wcel 1823   =/= wne 2649   A.wral 2804   E.wrex 2805   \ cdif 3458   i^i cin 3460   C_ wss 3461   (/)c0 3783   Ord word 4866   Oncon0 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565

This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 972  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-pss 3477  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-tp 4021  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-tr 4533  df-eprel 4780  df-po 4789  df-so 4790  df-fr 4827  df-we 4829  df-ord 4870  df-on 4871

This theorem is referenced by:  tfis  6662  tfisg  29527