Theorem fin0 6955

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 6829	. . 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 536	. . . . . . 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 4060	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
6		en0 6863	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
7	5, 6	sylib 122	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
8		nner 2371	. . . . 5 |- ( A = (/) -> -. A =/= (/) )
9	7, 8	syl 14	. . . 4 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A =/= (/) )
10		n0r 3465	. . . . . 6 |- ( E. x x e. A -> A =/= (/) )
11	10	necon2bi 2422	. . . . 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 703	. . 3 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A =/= (/) <-> E. x x e. A ) )
14		simplrr 536	. . . . . . . . . 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 6851	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
17		bren 6815	. . . . . . . 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 5507	. . . . . . . . . . . 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 4452	. . . . . . . . . . . . 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 528	. . . . . . . . . . . 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 2276	. . . . . . . . . . 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 5701	. . . . . . . . . 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 2779	. . . . . . . . . 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 1913	. . . . . 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 2614	. . . 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 4641	. . . 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 799	. 2 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( A =/= (/) <-> E. x x e. A ) )
37	2, 36	rexlimddv 2619	1 |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1506   e. wcel 2167   =/= wne 2367   E.wrex 2476   (/)c0 3451   class class class wbr 4034   suc csuc 4401   omcom 4627   -->wf 5255   -1-1-onto->wf1o 5258   ` cfv 5259   ~~ cen 6806   Fincfn 6808

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-iinf 4625

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-v 2765  df-sbc 2990  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-br 4035  df-opab 4096  df-id 4329  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-er 6601  df-en 6809  df-fin 6811

This theorem is referenced by:  findcard2  6959  findcard2s  6960  diffisn  6963  fimax2gtri  6971  elfi2  7047  elfir  7048  fiuni  7053  fifo  7055  4sqlem12  12596