Theorem findcard 7755

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 7536	. . 3 |- ( x e. Fin <-> E. w e. om x ~~ w )
3		breq2 4451	. . . . . . . 8 |- ( w = (/) -> ( x ~~ w <-> x ~~ (/) ) )
4	3	imbi1d 317	. . . . . . 7 |- ( w = (/) -> ( ( x ~~ w -> ph ) <-> ( x ~~ (/) -> ph ) ) )
5	4	albidv 1689	. . . . . 6 |- ( w = (/) -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ (/) -> ph ) ) )
6		breq2 4451	. . . . . . . 8 |- ( w = v -> ( x ~~ w <-> x ~~ v ) )
7	6	imbi1d 317	. . . . . . 7 |- ( w = v -> ( ( x ~~ w -> ph ) <-> ( x ~~ v -> ph ) ) )
8	7	albidv 1689	. . . . . 6 |- ( w = v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ v -> ph ) ) )
9		breq2 4451	. . . . . . . 8 |- ( w = suc v -> ( x ~~ w <-> x ~~ suc v ) )
10	9	imbi1d 317	. . . . . . 7 |- ( w = suc v -> ( ( x ~~ w -> ph ) <-> ( x ~~ suc v -> ph ) ) )
11	10	albidv 1689	. . . . . 6 |- ( w = suc v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ suc v -> ph ) ) )
12		en0 7575	. . . . . . . 8 |- ( x ~~ (/) <-> x = (/) )
13		findcard.5	. . . . . . . . 9 |- ps
14		findcard.1	. . . . . . . . 9 |- ( x = (/) -> ( ph <-> ps ) )
15	13, 14	mpbiri 233	. . . . . . . 8 |- ( x = (/) -> ph )
16	12, 15	sylbi 195	. . . . . . 7 |- ( x ~~ (/) -> ph )
17	16	ax-gen 1601	. . . . . 6 |- A. x ( x ~~ (/) -> ph )
18		peano2 6698	. . . . . . . . . . . . 13 |- ( v e. om -> suc v e. om )
19		breq2 4451	. . . . . . . . . . . . . 14 |- ( w = suc v -> ( y ~~ w <-> y ~~ suc v ) )
20	19	rspcev 3214	. . . . . . . . . . . . 13 |- ( ( suc v e. om /\ y ~~ suc v ) -> E. w e. om y ~~ w )
21	18, 20	sylan 471	. . . . . . . . . . . 12 |- ( ( v e. om /\ y ~~ suc v ) -> E. w e. om y ~~ w )
22		isfi 7536	. . . . . . . . . . . 12 |- ( y e. Fin <-> E. w e. om y ~~ w )
23	21, 22	sylibr 212	. . . . . . . . . . 11 |- ( ( v e. om /\ y ~~ suc v ) -> y e. Fin )
24	23	3adant2 1015	. . . . . . . . . 10 |- ( ( v e. om /\ A. x ( x ~~ v -> ph ) /\ y ~~ suc v ) -> y e. Fin )
25		dif1en 7749	. . . . . . . . . . . . . . . 16 |- ( ( v e. om /\ y ~~ suc v /\ z e. y ) -> ( y \ { z } ) ~~ v )
26	25	3expa 1196	. . . . . . . . . . . . . . 15 |- ( ( ( v e. om /\ y ~~ suc v ) /\ z e. y ) -> ( y \ { z } ) ~~ v )
27		vex 3116	. . . . . . . . . . . . . . . . 17 |- y e. _V
28		difexg 4595	. . . . . . . . . . . . . . . . 17 |- ( y e. _V -> ( y \ { z } ) e. _V )
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( y \ { z } ) e. _V
30		breq1 4450	. . . . . . . . . . . . . . . . 17 |- ( x = ( y \ { z } ) -> ( x ~~ v <-> ( y \ { z } ) ~~ v ) )
31		findcard.2	. . . . . . . . . . . . . . . . 17 |- ( x = ( y \ { z } ) -> ( ph <-> ch ) )
32	30, 31	imbi12d 320	. . . . . . . . . . . . . . . 16 |- ( x = ( y \ { z } ) -> ( ( x ~~ v -> ph ) <-> ( ( y \ { z } ) ~~ v -> ch ) ) )
33	29, 32	spcv 3204	. . . . . . . . . . . . . . 15 |- ( A. x ( x ~~ v -> ph ) -> ( ( y \ { z } ) ~~ v -> ch ) )
34	26, 33	syl5com 30	. . . . . . . . . . . . . 14 |- ( ( ( v e. om /\ y ~~ suc v ) /\ z e. y ) -> ( A. x ( x ~~ v -> ph ) -> ch ) )
35	34	ralrimdva 2882	. . . . . . . . . . . . 13 |- ( ( v e. om /\ y ~~ suc v ) -> ( A. x ( x ~~ v -> ph ) -> A. z e. y ch ) )
36	35	imp 429	. . . . . . . . . . . 12 |- ( ( ( v e. om /\ y ~~ suc v ) /\ A. x ( x ~~ v -> ph ) ) -> A. z e. y ch )
37	36	an32s 802	. . . . . . . . . . 11 |- ( ( ( v e. om /\ A. x ( x ~~ v -> ph ) ) /\ y ~~ suc v ) -> A. z e. y ch )
38	37	3impa 1191	. . . . . . . . . 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 60	. . . . . . . . 9 |- ( ( v e. om /\ A. x ( x ~~ v -> ph ) /\ y ~~ suc v ) -> th )
41	40	3exp 1195	. . . . . . . 8 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> ( y ~~ suc v -> th ) ) )
42	41	alrimdv 1697	. . . . . . 7 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> A. y ( y ~~ suc v -> th ) ) )
43		breq1 4450	. . . . . . . . 9 |- ( x = y -> ( x ~~ suc v <-> y ~~ suc v ) )
44		findcard.3	. . . . . . . . 9 |- ( x = y -> ( ph <-> th ) )
45	43, 44	imbi12d 320	. . . . . . . 8 |- ( x = y -> ( ( x ~~ suc v -> ph ) <-> ( y ~~ suc v -> th ) ) )
46	45	cbvalv 1996	. . . . . . 7 |- ( A. x ( x ~~ suc v -> ph ) <-> A. y ( y ~~ suc v -> th ) )
47	42, 46	syl6ibr 227	. . . . . 6 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> A. x ( x ~~ suc v -> ph ) ) )
48	5, 8, 11, 17, 47	finds1 6707	. . . . 5 |- ( w e. om -> A. x ( x ~~ w -> ph ) )
49	48	19.21bi 1818	. . . 4 |- ( w e. om -> ( x ~~ w -> ph ) )
50	49	rexlimiv 2949	. . 3 |- ( E. w e. om x ~~ w -> ph )
51	2, 50	sylbi 195	. 2 |- ( x e. Fin -> ph )
52	1, 51	vtoclga 3177	1 |- ( A e. Fin -> ta )

Syntax hints:   -> wi 4   <-> wb 184   /\ wa 369   /\ w3a 973   A.wal 1377   = wceq 1379   e. wcel 1767   A.wral 2814   E.wrex 2815   _Vcvv 3113   \ cdif 3473   (/)c0 3785   {csn 4027   class class class wbr 4447   suc csuc 4880   omcom 6678   ~~ cen 7510   Fincfn 7513

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-sbc 3332  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-om 6679  df-1o 7127  df-er 7308  df-en 7514  df-fin 7517

This theorem is referenced by:  xpfi  7787