Theorem fin0 7189

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 7047	. . 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 542	. . . . . . 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 4156	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
6		en0 7082	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
7	5, 6	sylib 122	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
8		nner 2424	. . . . 5 |- ( A = (/) -> -. A =/= (/) )
9	7, 8	syl 14	. . . 4 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A =/= (/) )
10		n0r 3535	. . . . . 6 |- ( E. x x e. A -> A =/= (/) )
11	10	necon2bi 2475	. . . . 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 714	. . 3 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> ( A =/= (/) <-> E. x x e. A ) )
14		simplrr 542	. . . . . . . . . 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 7070	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
17		bren 7030	. . . . . . . 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 5639	. . . . . . . . . . . 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 4561	. . . . . . . . . . . . 13 |- ( m e. om -> m e. suc m )
22	21	ad3antlr 497	. . . . . . . . . . . 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 533	. . . . . . . . . . . 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 2318	. . . . . . . . . . 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 5844	. . . . . . . . . 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 2838	. . . . . . . . . 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 1954	. . . . . 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 2668	. . . 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 4751	. . . 4 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
35	34	ad2antrl 494	. . 3 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) \/ E. m e. om n = suc m ) )
36	13, 33, 35	mpjaodan 810	. 2 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( A =/= (/) <-> E. x x e. A ) )
37	2, 36	rexlimddv 2673	1 |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720   = wceq 1402   E.wex 1545   e. wcel 2209   =/= wne 2420   E.wrex 2529   (/)c0 3520   class class class wbr 4130   suc csuc 4510   omcom 4737   -->wf 5373   -1-1-onto->wf1o 5376   ` cfv 5377   ~~ cen 7020   Fincfn 7022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-id 4438  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-er 6807  df-en 7023  df-fin 7025

This theorem is used by:  findcard2  7193  findcard2s  7194  diffisn  7197  fimax2gtri  7206  elfi2  7306  elfir  7307  fiuni  7312  fifo  7314  4sqlem12  13181