Theorem findcard 6944

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 6815	. . 3 |- ( x e. Fin <-> E. w e. om x ~~ w )
3		breq2 4033	. . . . . . . 8 |- ( w = (/) -> ( x ~~ w <-> x ~~ (/) ) )
4	3	imbi1d 231	. . . . . . 7 |- ( w = (/) -> ( ( x ~~ w -> ph ) <-> ( x ~~ (/) -> ph ) ) )
5	4	albidv 1835	. . . . . 6 |- ( w = (/) -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ (/) -> ph ) ) )
6		breq2 4033	. . . . . . . 8 |- ( w = v -> ( x ~~ w <-> x ~~ v ) )
7	6	imbi1d 231	. . . . . . 7 |- ( w = v -> ( ( x ~~ w -> ph ) <-> ( x ~~ v -> ph ) ) )
8	7	albidv 1835	. . . . . 6 |- ( w = v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ v -> ph ) ) )
9		breq2 4033	. . . . . . . 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 1835	. . . . . 6 |- ( w = suc v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ suc v -> ph ) ) )
12		en0 6849	. . . . . . . 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 1460	. . . . . 6 |- A. x ( x ~~ (/) -> ph )
18		peano2 4627	. . . . . . . . . . . . 13 |- ( v e. om -> suc v e. om )
19		breq2 4033	. . . . . . . . . . . . . 14 |- ( w = suc v -> ( y ~~ w <-> y ~~ suc v ) )
20	19	rspcev 2864	. . . . . . . . . . . . 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 6815	. . . . . . . . . . . 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 1018	. . . . . . . . . 10 |- ( ( v e. om /\ A. x ( x ~~ v -> ph ) /\ y ~~ suc v ) -> y e. Fin )
25		dif1en 6935	. . . . . . . . . . . . . . . 16 |- ( ( v e. om /\ y ~~ suc v /\ z e. y ) -> ( y \ { z } ) ~~ v )
26	25	3expa 1205	. . . . . . . . . . . . . . 15 |- ( ( ( v e. om /\ y ~~ suc v ) /\ z e. y ) -> ( y \ { z } ) ~~ v )
27		vex 2763	. . . . . . . . . . . . . . . . 17 |- y e. _V
28		difexg 4170	. . . . . . . . . . . . . . . . 17 |- ( y e. _V -> ( y \ { z } ) e. _V )
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( y \ { z } ) e. _V
30		breq1 4032	. . . . . . . . . . . . . . . . 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 2854	. . . . . . . . . . . . . . 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 2574	. . . . . . . . . . . . 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 1196	. . . . . . . . . 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 1204	. . . . . . . 8 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> ( y ~~ suc v -> th ) ) )
42	41	alrimdv 1887	. . . . . . 7 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> A. y ( y ~~ suc v -> th ) ) )
43		breq1 4032	. . . . . . . . 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 1929	. . . . . . 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 4634	. . . . 5 |- ( w e. om -> A. x ( x ~~ w -> ph ) )
49	48	19.21bi 1569	. . . 4 |- ( w e. om -> ( x ~~ w -> ph ) )
50	49	rexlimiv 2605	. . 3 |- ( E. w e. om x ~~ w -> ph )
51	2, 50	sylbi 121	. 2 |- ( x e. Fin -> ph )
52	1, 51	vtoclga 2826	1 |- ( A e. Fin -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 980   A.wal 1362   = wceq 1364   e. wcel 2164   A.wral 2472   E.wrex 2473   _Vcvv 2760   \ cdif 3150   (/)c0 3446   {csn 3618   class class class wbr 4029   suc csuc 4396   omcom 4622   ~~ cen 6792   Fincfn 6794

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-nul 4155  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-iinf 4620

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-if 3558  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-tr 4128  df-id 4324  df-iord 4397  df-on 4399  df-suc 4402  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-er 6587  df-en 6795  df-fin 6797

This theorem is referenced by:  xpfi  6986