Theorem ficard 10176

Description: A set is finite iff its cardinal is a natural number. (Contributed by Jeff Madsen 2-Sep-2009.)

Assertion

Ref	Expression
ficard	|- (A e. B -> (A e. Fin <-> (card` A) e. om))

Proof of Theorem ficard

Step	Hyp	Ref	Expression
1		carden 5981	. . . . 5 |- ((A e. B /\ x e. om) -> ((card` A) = (card` x) <-> A ~~ x))
2		cardnn 5870	. . . . . . . 8 |- (x e. om -> (card` x) = x)
3		eqtr 1904	. . . . . . . . 9 |- (((card` A) = (card` x) /\ (card` x) = x) -> (card` A) = x)
4	3	expcom 403	. . . . . . . 8 |- ((card` x) = x -> ((card` A) = (card` x) -> (card` A) = x))
5	2, 4	syl 12	. . . . . . 7 |- (x e. om -> ((card` A) = (card` x) -> (card` A) = x))
6		eleq1a 1966	. . . . . . 7 |- (x e. om -> ((card` A) = x -> (card` A) e. om))
7	5, 6	syld 30	. . . . . 6 |- (x e. om -> ((card` A) = (card` x) -> (card` A) e. om))
8	7	adantl 424	. . . . 5 |- ((A e. B /\ x e. om) -> ((card` A) = (card` x) -> (card` A) e. om))
9	1, 8	sylbird 222	. . . 4 |- ((A e. B /\ x e. om) -> (A ~~ x -> (card` A) e. om))
10	9	r19.23adva 2216	. . 3 |- (A e. B -> (E.x e. om A ~~ x -> (card` A) e. om))
11		isfi 5441	. . 3 |- (A e. Fin <-> E.x e. om A ~~ x)
12	10, 11	syl5ib 223	. 2 |- (A e. B -> (A e. Fin -> (card` A) e. om))
13		cardnn 5870	. . . . . . . 8 |- ((card` A) e. om -> (card` (card` A)) = (card` A))
14	13	eqcomd 1889	. . . . . . 7 |- ((card` A) e. om -> (card` A) = (card` (card` A)))
15	14	adantl 424	. . . . . 6 |- ((A e. B /\ (card` A) e. om) -> (card` A) = (card` (card` A)))
16		carden 5981	. . . . . 6 |- ((A e. B /\ (card` A) e. om) -> ((card` A) = (card` (card` A)) <-> A ~~ (card` A)))
17	15, 16	mpbid 212	. . . . 5 |- ((A e. B /\ (card` A) e. om) -> A ~~ (card` A))
18	17	ex 402	. . . 4 |- (A e. B -> ((card` A) e. om -> A ~~ (card` A)))
19	18	ancld 322	. . 3 |- (A e. B -> ((card` A) e. om -> ((card` A) e. om /\ A ~~ (card` A))))
20		breq2 3342	. . . . 5 |- (x = (card` A) -> (A ~~ x <-> A ~~ (card` A)))
21	20	rcla4ev 2381	. . . 4 |- (((card` A) e. om /\ A ~~ (card` A)) -> E.x e. om A ~~ x)
22	21, 11	sylibr 217	. . 3 |- (((card` A) e. om /\ A ~~ (card` A)) -> A e. Fin)
23	19, 22	syl6 25	. 2 |- (A e. B -> ((card` A) e. om -> A e. Fin))
24	12, 23	impbid 574	1 |- (A e. B -> (A e. Fin <-> (card` A) e. om))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  E.wrex 2106   class class class wbr 3338  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-er 5318  df-en 5427  df-dom 5428  df-sdom 5429  df-fin 5430  df-card 5862

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