Theorem findcard 7070

Description: Schema for induction on the cardinality of a finite set. The inductive hypothesis is that the result is true on the given set with any one element removed. The result is then proven to be true for all finite sets. (Contributed by Jeff Madsen, 2-Sep-2009.)

Hypotheses

Ref	Expression
findcard.1	|- ( x = (/) -> ( ph <-> ps ) )
findcard.2	|- ( x = ( y \ { z } ) -> ( ph <-> ch ) )
findcard.3	|- ( x = y -> ( ph <-> th ) )
findcard.4	|- ( x = A -> ( ph <-> ta ) )
findcard.5	|- ps
findcard.6	|- ( y e. Fin -> ( A. z e. y ch -> th ) )

Assertion

Ref	Expression
findcard	|- ( A e. Fin -> ta )

Distinct variable groups:   x, y, z, A   ps, x   ch, x   th, x   ta, x   ph, y, z

Allowed substitution hints:   ph( x)   ps( y, z)   ch( y, z)   th( y, z)   ta( y, z)

Proof of Theorem findcard

Dummy variables w v are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		findcard.4	. 2 |- ( x = A -> ( ph <-> ta ) )
2		isfi 6929	. . 3 |- ( x e. Fin <-> E. w e. om x ~~ w )
3		breq2 4090	. . . . . . . 8 |- ( w = (/) -> ( x ~~ w <-> x ~~ (/) ) )
4	3	imbi1d 231	. . . . . . 7 |- ( w = (/) -> ( ( x ~~ w -> ph ) <-> ( x ~~ (/) -> ph ) ) )
5	4	albidv 1870	. . . . . 6 |- ( w = (/) -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ (/) -> ph ) ) )
6		breq2 4090	. . . . . . . 8 |- ( w = v -> ( x ~~ w <-> x ~~ v ) )
7	6	imbi1d 231	. . . . . . 7 |- ( w = v -> ( ( x ~~ w -> ph ) <-> ( x ~~ v -> ph ) ) )
8	7	albidv 1870	. . . . . 6 |- ( w = v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ v -> ph ) ) )
9		breq2 4090	. . . . . . . 8 |- ( w = suc v -> ( x ~~ w <-> x ~~ suc v ) )
10	9	imbi1d 231	. . . . . . 7 |- ( w = suc v -> ( ( x ~~ w -> ph ) <-> ( x ~~ suc v -> ph ) ) )
11	10	albidv 1870	. . . . . 6 |- ( w = suc v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ suc v -> ph ) ) )
12		en0 6964	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
13		findcard.5	. . . . . . . . 9 |- ps
14		findcard.1	. . . . . . . . 9 |- ( x = (/) -> ( ph <-> ps ) )
15	13, 14	mpbiri 168	. . . . . . . 8 |- ( x = (/) -> ph )
16	12, 15	sylbi 121	. . . . . . 7 |- ( x ~~ (/) -> ph )
17	16	ax-gen 1495	. . . . . 6 |- A. x ( x ~~ (/) -> ph )
18		peano2 4691	. . . . . . . . . . . . 13 |- ( v e. om -> suc v e. om )
19		breq2 4090	. . . . . . . . . . . . . 14 |- ( w = suc v -> ( y ~~ w <-> y ~~ suc v ) )
20	19	rspcev 2908	. . . . . . . . . . . . 13 |- ( ( suc v e. om /\ y ~~ suc v ) -> E. w e. om y ~~ w )
21	18, 20	sylan 283	. . . . . . . . . . . 12 |- ( ( v e. om /\ y ~~ suc v ) -> E. w e. om y ~~ w )
22		isfi 6929	. . . . . . . . . . . 12 |- ( y e. Fin <-> E. w e. om y ~~ w )
23	21, 22	sylibr 134	. . . . . . . . . . 11 |- ( ( v e. om /\ y ~~ suc v ) -> y e. Fin )
24	23	3adant2 1040	. . . . . . . . . 10 |- ( ( v e. om /\ A. x ( x ~~ v -> ph ) /\ y ~~ suc v ) -> y e. Fin )
25		dif1en 7061	. . . . . . . . . . . . . . . 16 |- ( ( v e. om /\ y ~~ suc v /\ z e. y ) -> ( y \ { z } ) ~~ v )
26	25	3expa 1227	. . . . . . . . . . . . . . 15 |- ( ( ( v e. om /\ y ~~ suc v ) /\ z e. y ) -> ( y \ { z } ) ~~ v )
27		vex 2803	. . . . . . . . . . . . . . . . 17 |- y e. _V
28		difexg 4229	. . . . . . . . . . . . . . . . 17 |- ( y e. _V -> ( y \ { z } ) e. _V )
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( y \ { z } ) e. _V
30		breq1 4089	. . . . . . . . . . . . . . . . 17 |- ( x = ( y \ { z } ) -> ( x ~~ v <-> ( y \ { z } ) ~~ v ) )
31		findcard.2	. . . . . . . . . . . . . . . . 17 |- ( x = ( y \ { z } ) -> ( ph <-> ch ) )
32	30, 31	imbi12d 234	. . . . . . . . . . . . . . . 16 |- ( x = ( y \ { z } ) -> ( ( x ~~ v -> ph ) <-> ( ( y \ { z } ) ~~ v -> ch ) ) )
33	29, 32	spcv 2898	. . . . . . . . . . . . . . 15 |- ( A. x ( x ~~ v -> ph ) -> ( ( y \ { z } ) ~~ v -> ch ) )
34	26, 33	syl5com 29	. . . . . . . . . . . . . 14 |- ( ( ( v e. om /\ y ~~ suc v ) /\ z e. y ) -> ( A. x ( x ~~ v -> ph ) -> ch ) )
35	34	ralrimdva 2610	. . . . . . . . . . . . 13 |- ( ( v e. om /\ y ~~ suc v ) -> ( A. x ( x ~~ v -> ph ) -> A. z e. y ch ) )
36	35	imp 124	. . . . . . . . . . . 12 |- ( ( ( v e. om /\ y ~~ suc v ) /\ A. x ( x ~~ v -> ph ) ) -> A. z e. y ch )
37	36	an32s 568	. . . . . . . . . . 11 |- ( ( ( v e. om /\ A. x ( x ~~ v -> ph ) ) /\ y ~~ suc v ) -> A. z e. y ch )
38	37	3impa 1218	. . . . . . . . . 10 |- ( ( v e. om /\ A. x ( x ~~ v -> ph ) /\ y ~~ suc v ) -> A. z e. y ch )
39		findcard.6	. . . . . . . . . 10 |- ( y e. Fin -> ( A. z e. y ch -> th ) )
40	24, 38, 39	sylc 62	. . . . . . . . 9 |- ( ( v e. om /\ A. x ( x ~~ v -> ph ) /\ y ~~ suc v ) -> th )
41	40	3exp 1226	. . . . . . . 8 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> ( y ~~ suc v -> th ) ) )
42	41	alrimdv 1922	. . . . . . 7 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> A. y ( y ~~ suc v -> th ) ) )
43		breq1 4089	. . . . . . . . 9 |- ( x = y -> ( x ~~ suc v <-> y ~~ suc v ) )
44		findcard.3	. . . . . . . . 9 |- ( x = y -> ( ph <-> th ) )
45	43, 44	imbi12d 234	. . . . . . . 8 |- ( x = y -> ( ( x ~~ suc v -> ph ) <-> ( y ~~ suc v -> th ) ) )
46	45	cbvalv 1964	. . . . . . 7 |- ( A. x ( x ~~ suc v -> ph ) <-> A. y ( y ~~ suc v -> th ) )
47	42, 46	imbitrrdi 162	. . . . . 6 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> A. x ( x ~~ suc v -> ph ) ) )
48	5, 8, 11, 17, 47	finds1 4698	. . . . 5 |- ( w e. om -> A. x ( x ~~ w -> ph ) )
49	48	19.21bi 1604	. . . 4 |- ( w e. om -> ( x ~~ w -> ph ) )
50	49	rexlimiv 2642	. . 3 |- ( E. w e. om x ~~ w -> ph )
51	2, 50	sylbi 121	. 2 |- ( x e. Fin -> ph )
52	1, 51	vtoclga 2868	1 |- ( A e. Fin -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 1002   A.wal 1393   = wceq 1395   e. wcel 2200   A.wral 2508   E.wrex 2509   _Vcvv 2800   \ cdif 3195   (/)c0 3492   {csn 3667   class class class wbr 4086   suc csuc 4460   omcom 4686   ~~ cen 6902   Fincfn 6904

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-if 3604  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-er 6697  df-en 6905  df-fin 6907

This theorem is referenced by:  xpfi  7117