Theorem tfinds2 6698

Description: Transfinite Induction (inference schema), using implicit substitutions. The first three hypotheses establish the substitutions we need. The last three are the basis and the induction hypotheses (for successor and limit ordinals respectively). Theorem Schema 4 of [Suppes] p. 197. The wff is an auxiliary antecedent to help shorten proofs using this theorem. (Contributed by NM, 4-Sep-2004.)

Hypotheses

Ref	Expression
tfinds2.1
tfinds2.2
tfinds2.3
tfinds2.4
tfinds2.5
tfinds2.6

Assertion

Ref	Expression
tfinds2

Distinct variable groups:   ,,   ,   ,   ,   ,

Proof of Theorem tfinds2

Step	Hyp	Ref	Expression
1		tfinds2.4	. . 3
2		0ex 4582	. . . 4
3		tfinds2.1	. . . . 5
4	3	imbi2d 316	. . . 4
5	2, 4	sbcie 3362	. . 3
6	1, 5	mpbir 209	. 2
7		vex 3112	. . . . . 6
8		tfinds2.5	. . . . . . . 8
9	8	a2d 26	. . . . . . 7
10	9	sbcth 3342	. . . . . 6
11	7, 10	ax-mp 5	. . . . 5
12		sbcimg 3369	. . . . . 6
13	7, 12	ax-mp 5	. . . . 5
14	11, 13	mpbi 208	. . . 4
15		sbcel1v 3392	. . . 4
16		sbcimg 3369	. . . . 5
17	7, 16	ax-mp 5	. . . 4
18	14, 15, 17	3imtr3i 265	. . 3
19		tfinds2.2	. . . . . . 7
20	19	bicomd 201	. . . . . 6
21	20	equcoms 1795	. . . . 5
22	21	imbi2d 316	. . . 4
23	7, 22	sbcie 3362	. . 3
24		vex 3112	. . . . . . 7
25	24	sucex 6646	. . . . . 6
26		tfinds2.3	. . . . . . 7
27	26	imbi2d 316	. . . . . 6
28	25, 27	sbcie 3362	. . . . 5
29	28	sbcbii 3387	. . . 4
30		suceq 4948	. . . . 5
31	30	sbcco2 3351	. . . 4
32	29, 31	bitr3i 251	. . 3
33	18, 23, 32	3imtr3g 269	. 2
34		sbsbc 3331	. . . 4
35	22	sbralie 3097	. . . 4
36	34, 35	bitr3i 251	. . 3
37		r19.21v 2862	. . . . . . . 8
38		tfinds2.6	. . . . . . . . 9
39	38	a2d 26	. . . . . . . 8
40	37, 39	syl5bi 217	. . . . . . 7
41	40	sbcth 3342	. . . . . 6
42	24, 41	ax-mp 5	. . . . 5
43		sbcimg 3369	. . . . . 6
44	24, 43	ax-mp 5	. . . . 5
45	42, 44	mpbi 208	. . . 4
46		limeq 4895	. . . . 5
47	24, 46	sbcie 3362	. . . 4
48		sbcimg 3369	. . . . 5
49	24, 48	ax-mp 5	. . . 4
50	45, 47, 49	3imtr3i 265	. . 3
51	36, 50	syl5bir 218	. 2
52	6, 33, 51	tfindes 6697	1

Syntax hints:  ->wi 4  <->wb 184  =wceq 1395  [wsb 1739  e.wcel 1818  A.wral 2807  cvv 3109  [.wsbc 3327  c0 3784  con0 4883  Limwlim 4884  succsuc 4885

This theorem is referenced by:  inar1  9174  grur1a  9218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-8 1820  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691  ax-un 6592

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-pss 3491  df-nul 3785  df-if 3942  df-pw 4014  df-sn 4030  df-pr 4032  df-tp 4034  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-tr 4546  df-eprel 4796  df-po 4805  df-so 4806  df-fr 4843  df-we 4845  df-ord 4886  df-on 4887  df-lim 4888  df-suc 4889