Theorem dif1card 10177

Description: The cardinality of a non-empty finite set is one greater than the cardinality of the set with one element removed. (Contributed by Jeff Madsen 2-Sep-2009.)

Assertion

Ref	Expression
dif1card	|- ((A e. Fin /\ X e. A) -> (card` A) = suc (card` (A \ {X})))

Proof of Theorem dif1card

Step	Hyp	Ref	Expression
1		isfi 5441	. . . 4 |- (A e. Fin <-> E.x e. om A ~~ x)
2	1	biimpi 168	. . 3 |- (A e. Fin -> E.x e. om A ~~ x)
3	2	adantr 425	. 2 |- ((A e. Fin /\ X e. A) -> E.x e. om A ~~ x)
4		n0i 2880	. . . . 5 |- (X e. A -> -. A = (/))
5		nn0suc 3976	. . . . . . . 8 |- (x e. om -> (x = (/) \/ E.y e. om x = suc y))
6		breq2 3342	. . . . . . . . . . . . 13 |- (x = (/) -> (A ~~ x <-> A ~~ (/)))
7	6	biimpd 170	. . . . . . . . . . . 12 |- (x = (/) -> (A ~~ x -> A ~~ (/)))
8		en0 5482	. . . . . . . . . . . 12 |- (A ~~ (/) <-> A = (/))
9	7, 8	syl6ib 229	. . . . . . . . . . 11 |- (x = (/) -> (A ~~ x -> A = (/)))
10		pm2.24 95	. . . . . . . . . . 11 |- (A = (/) -> (-. A = (/) -> E.y e. om A ~~ suc y))
11	9, 10	syl6 25	. . . . . . . . . 10 |- (x = (/) -> (A ~~ x -> (-. A = (/) -> E.y e. om A ~~ suc y)))
12	11	com23 36	. . . . . . . . 9 |- (x = (/) -> (-. A = (/) -> (A ~~ x -> E.y e. om A ~~ suc y)))
13		breq2 3342	. . . . . . . . . . . . 13 |- (x = suc y -> (A ~~ x <-> A ~~ suc y))
14	13	biimpcd 172	. . . . . . . . . . . 12 |- (A ~~ x -> (x = suc y -> A ~~ suc y))
15	14	reximdv 2202	. . . . . . . . . . 11 |- (A ~~ x -> (E.y e. om x = suc y -> E.y e. om A ~~ suc y))
16	15	com12 14	. . . . . . . . . 10 |- (E.y e. om x = suc y -> (A ~~ x -> E.y e. om A ~~ suc y))
17	16	a1d 15	. . . . . . . . 9 |- (E.y e. om x = suc y -> (-. A = (/) -> (A ~~ x -> E.y e. om A ~~ suc y)))
18	12, 17	jaoi 368	. . . . . . . 8 |- ((x = (/) \/ E.y e. om x = suc y) -> (-. A = (/) -> (A ~~ x -> E.y e. om A ~~ suc y)))
19	5, 18	syl 12	. . . . . . 7 |- (x e. om -> (-. A = (/) -> (A ~~ x -> E.y e. om A ~~ suc y)))
20	19	impcom 378	. . . . . 6 |- ((-. A = (/) /\ x e. om) -> (A ~~ x -> E.y e. om A ~~ suc y))
21	20	r19.23adva 2216	. . . . 5 |- (-. A = (/) -> (E.x e. om A ~~ x -> E.y e. om A ~~ suc y))
22	4, 21	syl 12	. . . 4 |- (X e. A -> (E.x e. om A ~~ x -> E.y e. om A ~~ suc y))
23	22	adantl 424	. . 3 |- ((A e. Fin /\ X e. A) -> (E.x e. om A ~~ x -> E.y e. om A ~~ suc y))
24		elisset 2299	. . . . . . . . . . . 12 |- (A e. Fin -> A e. _V)
25		breq1 3341	. . . . . . . . . . . . . . 15 |- (A = if(A e. _V, A, (/)) -> (A ~~ suc y <-> if(A e. _V, A, (/)) ~~ suc y))
26		eleq2 1958	. . . . . . . . . . . . . . 15 |- (A = if(A e. _V, A, (/)) -> (X e. A <-> X e. if(A e. _V, A, (/))))
27	25, 26	3anbi23d 1171	. . . . . . . . . . . . . 14 |- (A = if(A e. _V, A, (/)) -> ((y e. om /\ A ~~ suc y /\ X e. A) <-> (y e. om /\ if(A e. _V, A, (/)) ~~ suc y /\ X e. if(A e. _V, A, (/)))))
28		difeq1 2717	. . . . . . . . . . . . . . 15 |- (A = if(A e. _V, A, (/)) -> (A \ {X}) = (if(A e. _V, A, (/)) \ {X}))
29	28	breq1d 3348	. . . . . . . . . . . . . 14 |- (A = if(A e. _V, A, (/)) -> ((A \ {X}) ~~ y <-> (if(A e. _V, A, (/)) \ {X}) ~~ y))
30	27, 29	imbi12d 688	. . . . . . . . . . . . 13 |- (A = if(A e. _V, A, (/)) -> (((y e. om /\ A ~~ suc y /\ X e. A) -> (A \ {X}) ~~ y) <-> ((y e. om /\ if(A e. _V, A, (/)) ~~ suc y /\ X e. if(A e. _V, A, (/))) -> (if(A e. _V, A, (/)) \ {X}) ~~ y)))
31		0ex 3446	. . . . . . . . . . . . . . 15 |- (/) e. _V
32	31	elimel 3025	. . . . . . . . . . . . . 14 |- if(A e. _V, A, (/)) e. _V
33	32	dif1en 10172	. . . . . . . . . . . . 13 |- ((y e. om /\ if(A e. _V, A, (/)) ~~ suc y /\ X e. if(A e. _V, A, (/))) -> (if(A e. _V, A, (/)) \ {X}) ~~ y)
34	30, 33	dedth 3011	. . . . . . . . . . . 12 |- (A e. _V -> ((y e. om /\ A ~~ suc y /\ X e. A) -> (A \ {X}) ~~ y))
35	24, 34	syl 12	. . . . . . . . . . 11 |- (A e. Fin -> ((y e. om /\ A ~~ suc y /\ X e. A) -> (A \ {X}) ~~ y))
36		cardennn 10171	. . . . . . . . . . . . . . . . 17 |- ((A e. Fin /\ A ~~ suc y /\ suc y e. om) -> (card` A) = suc y)
37		peano2 3972	. . . . . . . . . . . . . . . . 17 |- (y e. om -> suc y e. om)
38	36, 37	syl3an3 1132	. . . . . . . . . . . . . . . 16 |- ((A e. Fin /\ A ~~ suc y /\ y e. om) -> (card` A) = suc y)
39		cardennn 10171	. . . . . . . . . . . . . . . . . . . . . . 23 |- (((A \ {X}) e. _V /\ (A \ {X}) ~~ y /\ y e. om) -> (card` (A \ {X})) = y)
40		difexg 3458	. . . . . . . . . . . . . . . . . . . . . . 23 |- (A e. Fin -> (A \ {X}) e. _V)
41	39, 40	syl3an1 1130	. . . . . . . . . . . . . . . . . . . . . 22 |- ((A e. Fin /\ (A \ {X}) ~~ y /\ y e. om) -> (card` (A \ {X})) = y)
42	41	3exp 1066	. . . . . . . . . . . . . . . . . . . . 21 |- (A e. Fin -> ((A \ {X}) ~~ y -> (y e. om -> (card` (A \ {X})) = y)))
43	42	com23 36	. . . . . . . . . . . . . . . . . . . 20 |- (A e. Fin -> (y e. om -> ((A \ {X}) ~~ y -> (card` (A \ {X})) = y)))
44	43	imp 377	. . . . . . . . . . . . . . . . . . 19 |- ((A e. Fin /\ y e. om) -> ((A \ {X}) ~~ y -> (card` (A \ {X})) = y))
45		suceq 3729	. . . . . . . . . . . . . . . . . . . . 21 |- ((card` (A \ {X})) = y -> suc (card` (A \ {X})) = suc y)
46	45	eqeq2d 1895	. . . . . . . . . . . . . . . . . . . 20 |- ((card` (A \ {X})) = y -> ((card` A) = suc (card` (A \ {X})) <-> (card` A) = suc y))
47	46	biimprd 171	. . . . . . . . . . . . . . . . . . 19 |- ((card` (A \ {X})) = y -> ((card` A) = suc y -> (card` A) = suc (card` (A \ {X}))))
48	44, 47	syl6 25	. . . . . . . . . . . . . . . . . 18 |- ((A e. Fin /\ y e. om) -> ((A \ {X}) ~~ y -> ((card` A) = suc y -> (card` A) = suc (card` (A \ {X})))))
49	48	com23 36	. . . . . . . . . . . . . . . . 17 |- ((A e. Fin /\ y e. om) -> ((card` A) = suc y -> ((A \ {X}) ~~ y -> (card` A) = suc (card` (A \ {X})))))
50	49	3adant2 895	. . . . . . . . . . . . . . . 16 |- ((A e. Fin /\ A ~~ suc y /\ y e. om) -> ((card` A) = suc y -> ((A \ {X}) ~~ y -> (card` A) = suc (card` (A \ {X})))))
51	38, 50	mpd 29	. . . . . . . . . . . . . . 15 |- ((A e. Fin /\ A ~~ suc y /\ y e. om) -> ((A \ {X}) ~~ y -> (card` A) = suc (card` (A \ {X}))))
52	51	a1d 15	. . . . . . . . . . . . . 14 |- ((A e. Fin /\ A ~~ suc y /\ y e. om) -> (X e. A -> ((A \ {X}) ~~ y -> (card` A) = suc (card` (A \ {X})))))
53	52	3exp 1066	. . . . . . . . . . . . 13 |- (A e. Fin -> (A ~~ suc y -> (y e. om -> (X e. A -> ((A \ {X}) ~~ y -> (card` A) = suc (card` (A \ {X})))))))
54	53	com23 36	. . . . . . . . . . . 12 |- (A e. Fin -> (y e. om -> (A ~~ suc y -> (X e. A -> ((A \ {X}) ~~ y -> (card` A) = suc (card` (A \ {X})))))))
55	54	3impd 1082	. . . . . . . . . . 11 |- (A e. Fin -> ((y e. om /\ A ~~ suc y /\ X e. A) -> ((A \ {X}) ~~ y -> (card` A) = suc (card` (A \ {X})))))
56	35, 55	mpdd 57	. . . . . . . . . 10 |- (A e. Fin -> ((y e. om /\ A ~~ suc y /\ X e. A) -> (card` A) = suc (card` (A \ {X}))))
57	56	com12 14	. . . . . . . . 9 |- ((y e. om /\ A ~~ suc y /\ X e. A) -> (A e. Fin -> (card` A) = suc (card` (A \ {X}))))
58	57	3comr 1076	. . . . . . . 8 |- ((X e. A /\ y e. om /\ A ~~ suc y) -> (A e. Fin -> (card` A) = suc (card` (A \ {X}))))
59	58	3expia 1069	. . . . . . 7 |- ((X e. A /\ y e. om) -> (A ~~ suc y -> (A e. Fin -> (card` A) = suc (card` (A \ {X})))))
60	59	com23 36	. . . . . 6 |- ((X e. A /\ y e. om) -> (A e. Fin -> (A ~~ suc y -> (card` A) = suc (card` (A \ {X})))))
61	60	impcom 378	. . . . 5 |- ((A e. Fin /\ (X e. A /\ y e. om)) -> (A ~~ suc y -> (card` A) = suc (card` (A \ {X}))))
62	61	anassrs 489	. . . 4 |- (((A e. Fin /\ X e. A) /\ y e. om) -> (A ~~ suc y -> (card` A) = suc (card` (A \ {X}))))
63	62	r19.23adva 2216	. . 3 |- ((A e. Fin /\ X e. A) -> (E.y e. om A ~~ suc y -> (card` A) = suc (card` (A \ {X}))))
64	23, 63	syld 30	. 2 |- ((A e. Fin /\ X e. A) -> (E.x e. om A ~~ x -> (card` A) = suc (card` (A \ {X}))))
65	3, 64	mpd 29	1 |- ((A e. Fin /\ X e. A) -> (card` A) = suc (card` (A \ {X})))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  E.wrex 2106  _Vcvv 2292   \ cdif 2590  (/)c0 2875  ifcif 2982  {csn 3044   class class class wbr 3338  suc csuc 3659  omcom 3949  ` cfv 3998   ~~ cen 5423  Fincfn 5426  cardccrd 5859

This theorem is referenced by:  findcardOLD 10179

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790  ax-ac 5906

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3or 859  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-reu 2111  df-rab 2112  df-v 2294  df-sbc 2454  df-csb 2541  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876  df-if 2983  df-pw 3035  df-sn 3049  df-pr 3050  df-tp 3052  df-op 3053  df-uni 3178  df-int 3215  df-iun 3257  df-br 3339  df-opab 3396  df-tr 3412  df-eprel 3583  df-id 3586  df-po 3591  df-so 3604  df-fr 3625  df-we 3644  df-ord 3660  df-on 3661  df-lim 3662  df-suc 3663  df-om 3950  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-fv 4014  df-1o 5177  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862

Copyright terms: Public domain