Theorem fin0 7073

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 6933	. . 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 4114	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
6		en0 6968	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
7	5, 6	sylib 122	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
8		nner 2406	. . . . 5 |- ( A = (/) -> -. A =/= (/) )
9	7, 8	syl 14	. . . 4 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A =/= (/) )
10		n0r 3508	. . . . . 6 |- ( E. x x e. A -> A =/= (/) )
11	10	necon2bi 2457	. . . . 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 709	. . 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 6956	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
17		bren 6916	. . . . . . . 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 5583	. . . . . . . . . . . 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 4513	. . . . . . . . . . . . 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 2311	. . . . . . . . . . 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 5783	. . . . . . . . . 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 2819	. . . . . . . . . 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 1947	. . . . . 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 2650	. . . 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 4702	. . . 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 805	. 2 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( A =/= (/) <-> E. x x e. A ) )
37	2, 36	rexlimddv 2655	1 |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715   = wceq 1397   E.wex 1540   e. wcel 2202   =/= wne 2402   E.wrex 2511   (/)c0 3494   class class class wbr 4088   suc csuc 4462   omcom 4688   -->wf 5322   -1-1-onto->wf1o 5325   ` cfv 5326   ~~ cen 6906   Fincfn 6908

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-er 6701  df-en 6909  df-fin 6911

This theorem is referenced by:  findcard2  7077  findcard2s  7078  diffisn  7081  fimax2gtri  7090  elfi2  7170  elfir  7171  fiuni  7176  fifo  7178  4sqlem12  12974