Theorem fin0 7142

Description: A nonempty finite set has at least one element. (Contributed by Jim Kingdon, 10-Sep-2021.)

Assertion

Ref	Expression
fin0	|- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )

Distinct variable group:   x, A

Proof of Theorem fin0

Dummy variables f m n are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 7000	. . 3 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 120	. 2 |- ( A e. Fin -> E. n e. om A ~~ n )
3		simplrr 538	. . . . . . 7 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
4		simpr 110	. . . . . . 7 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> n = (/) )
5	3, 4	breqtrd 4135	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
6		en0 7035	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
7	5, 6	sylib 122	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
8		nner 2416	. . . . 5 |- ( A = (/) -> -. A =/= (/) )
9	7, 8	syl 14	. . . 4 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A =/= (/) )
10		n0r 3522	. . . . . 6 |- ( E. x x e. A -> A =/= (/) )
11	10	necon2bi 2467	. . . . 5 |- ( A = (/) -> -. E. x x e. A )
12	7, 11	syl 14	. . . 4 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. E. x x e. A )
13	9, 12	2falsed 710	. . 3 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A =/= (/) <-> E. x x e. A ) )
14		simplrr 538	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> A ~~ n )
15	14	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ n )
16	15	ensymd 7023	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
17		bren 6983	. . . . . . . 8 |- ( n ~~ A <-> E. f f : n -1-1-onto-> A )
18	16, 17	sylib 122	. . . . . . 7 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> E. f f : n -1-1-onto-> A )
19		f1of 5614	. . . . . . . . . . . 12 |- ( f : n -1-1-onto-> A -> f : n --> A )
20	19	adantl 277	. . . . . . . . . . 11 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> f : n --> A )
21		sucidg 4537	. . . . . . . . . . . . 13 |- ( m e. om -> m e. suc m )
22	21	ad3antlr 493	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> m e. suc m )
23		simplr 529	. . . . . . . . . . . 12 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> n = suc m )
24	22, 23	eleqtrrd 2312	. . . . . . . . . . 11 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> m e. n )
25	20, 24	ffvelcdmd 5813	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> ( f ` m ) e. A )
26		elex2 2830	. . . . . . . . . 10 |- ( ( f ` m ) e. A -> E. x x e. A )
27	25, 26	syl 14	. . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> E. x x e. A )
28	27, 10	syl 14	. . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> A =/= (/) )
29	28, 27	2thd 175	. . . . . . 7 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> ( A =/= (/) <-> E. x x e. A ) )
30	18, 29	exlimddv 1948	. . . . . 6 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> ( A =/= (/) <-> E. x x e. A ) )
31	30	ex 115	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> ( n = suc m -> ( A =/= (/) <-> E. x x e. A ) ) )
32	31	rexlimdva 2660	. . . 4 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( E. m e. om n = suc m -> ( A =/= (/) <-> E. x x e. A ) ) )
33	32	imp 124	. . 3 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ E. m e. om n = suc m ) -> ( A =/= (/) <-> E. x x e. A ) )
34		nn0suc 4726	. . . 4 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
35	34	ad2antrl 490	. . 3 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) \/ E. m e. om n = suc m ) )
36	13, 33, 35	mpjaodan 806	. 2 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( A =/= (/) <-> E. x x e. A ) )
37	2, 36	rexlimddv 2665	1 |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2203   =/= wne 2412   E.wrex 2521   (/)c0 3508   class class class wbr 4109   suc csuc 4486   omcom 4712   -->wf 5348   -1-1-onto->wf1o 5351   ` cfv 5352   ~~ cen 6973   Fincfn 6975

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-id 4414  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-er 6767  df-en 6976  df-fin 6978

This theorem is referenced by:  findcard2  7146  findcard2s  7147  diffisn  7150  fimax2gtri  7159  elfi2  7259  elfir  7260  fiuni  7265  fifo  7267  4sqlem12  13100