Theorem findcard 7050

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 6912	. . 3 |- ( x e. Fin <-> E. w e. om x ~~ w )
3		breq2 4087	. . . . . . . 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 4087	. . . . . . . 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 4087	. . . . . . . 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 6947	. . . . . . . 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 4687	. . . . . . . . . . . . 13 |- ( v e. om -> suc v e. om )
19		breq2 4087	. . . . . . . . . . . . . 14 |- ( w = suc v -> ( y ~~ w <-> y ~~ suc v ) )
20	19	rspcev 2907	. . . . . . . . . . . . 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 6912	. . . . . . . . . . . 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 7041	. . . . . . . . . . . . . . . 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 2802	. . . . . . . . . . . . . . . . 17 |- y e. _V
28		difexg 4225	. . . . . . . . . . . . . . . . 17 |- ( y e. _V -> ( y \ { z } ) e. _V )
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( y \ { z } ) e. _V
30		breq1 4086	. . . . . . . . . . . . . . . . 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 2897	. . . . . . . . . . . . . . 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 4086	. . . . . . . . 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 4694	. . . . 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 2867	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 2799   \ cdif 3194   (/)c0 3491   {csn 3666   class class class wbr 4083   suc csuc 4456   omcom 4682   ~~ cen 6885   Fincfn 6887

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 4199  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-if 3603  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-iun 3967  df-br 4084  df-opab 4146  df-mpt 4147  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-er 6680  df-en 6888  df-fin 6890

This theorem is referenced by:  xpfi  7094