Theorem findcard 7011

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 6875	. . 3 |- ( x e. Fin <-> E. w e. om x ~~ w )
3		breq2 4063	. . . . . . . 8 |- ( w = (/) -> ( x ~~ w <-> x ~~ (/) ) )
4	3	imbi1d 231	. . . . . . 7 |- ( w = (/) -> ( ( x ~~ w -> ph ) <-> ( x ~~ (/) -> ph ) ) )
5	4	albidv 1848	. . . . . 6 |- ( w = (/) -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ (/) -> ph ) ) )
6		breq2 4063	. . . . . . . 8 |- ( w = v -> ( x ~~ w <-> x ~~ v ) )
7	6	imbi1d 231	. . . . . . 7 |- ( w = v -> ( ( x ~~ w -> ph ) <-> ( x ~~ v -> ph ) ) )
8	7	albidv 1848	. . . . . 6 |- ( w = v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ v -> ph ) ) )
9		breq2 4063	. . . . . . . 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 1848	. . . . . 6 |- ( w = suc v -> ( A. x ( x ~~ w -> ph ) <-> A. x ( x ~~ suc v -> ph ) ) )
12		en0 6910	. . . . . . . 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 1473	. . . . . 6 |- A. x ( x ~~ (/) -> ph )
18		peano2 4661	. . . . . . . . . . . . 13 |- ( v e. om -> suc v e. om )
19		breq2 4063	. . . . . . . . . . . . . 14 |- ( w = suc v -> ( y ~~ w <-> y ~~ suc v ) )
20	19	rspcev 2884	. . . . . . . . . . . . 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 6875	. . . . . . . . . . . 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 1019	. . . . . . . . . 10 |- ( ( v e. om /\ A. x ( x ~~ v -> ph ) /\ y ~~ suc v ) -> y e. Fin )
25		dif1en 7002	. . . . . . . . . . . . . . . 16 |- ( ( v e. om /\ y ~~ suc v /\ z e. y ) -> ( y \ { z } ) ~~ v )
26	25	3expa 1206	. . . . . . . . . . . . . . 15 |- ( ( ( v e. om /\ y ~~ suc v ) /\ z e. y ) -> ( y \ { z } ) ~~ v )
27		vex 2779	. . . . . . . . . . . . . . . . 17 |- y e. _V
28		difexg 4201	. . . . . . . . . . . . . . . . 17 |- ( y e. _V -> ( y \ { z } ) e. _V )
29	27, 28	ax-mp 5	. . . . . . . . . . . . . . . 16 |- ( y \ { z } ) e. _V
30		breq1 4062	. . . . . . . . . . . . . . . . 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 2874	. . . . . . . . . . . . . . 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 2588	. . . . . . . . . . . . 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 1197	. . . . . . . . . 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 1205	. . . . . . . 8 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> ( y ~~ suc v -> th ) ) )
42	41	alrimdv 1900	. . . . . . 7 |- ( v e. om -> ( A. x ( x ~~ v -> ph ) -> A. y ( y ~~ suc v -> th ) ) )
43		breq1 4062	. . . . . . . . 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 1942	. . . . . . 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 4668	. . . . 5 |- ( w e. om -> A. x ( x ~~ w -> ph ) )
49	48	19.21bi 1582	. . . 4 |- ( w e. om -> ( x ~~ w -> ph ) )
50	49	rexlimiv 2619	. . 3 |- ( E. w e. om x ~~ w -> ph )
51	2, 50	sylbi 121	. 2 |- ( x e. Fin -> ph )
52	1, 51	vtoclga 2844	1 |- ( A e. Fin -> ta )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   /\ w3a 981   A.wal 1371   = wceq 1373   e. wcel 2178   A.wral 2486   E.wrex 2487   _Vcvv 2776   \ cdif 3171   (/)c0 3468   {csn 3643   class class class wbr 4059   suc csuc 4430   omcom 4656   ~~ cen 6848   Fincfn 6850

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-coll 4175  ax-sep 4178  ax-nul 4186  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-iinf 4654

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-reu 2493  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-nul 3469  df-if 3580  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-int 3900  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-tr 4159  df-id 4358  df-iord 4431  df-on 4433  df-suc 4436  df-iom 4657  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-er 6643  df-en 6851  df-fin 6853

This theorem is referenced by:  xpfi  7055