Theorem fin0 6982

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 6852	. . 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 4070	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
6		en0 6887	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
7	5, 6	sylib 122	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
8		nner 2380	. . . . 5 |- ( A = (/) -> -. A =/= (/) )
9	7, 8	syl 14	. . . 4 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> -. A =/= (/) )
10		n0r 3474	. . . . . 6 |- ( E. x x e. A -> A =/= (/) )
11	10	necon2bi 2431	. . . . 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 704	. . 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 6875	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
17		bren 6835	. . . . . . . 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 5522	. . . . . . . . . . . 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 4463	. . . . . . . . . . . . 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 2285	. . . . . . . . . . 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 5716	. . . . . . . . . 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 2788	. . . . . . . . . 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 1922	. . . . . 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 2623	. . . 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 4652	. . . 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 800	. 2 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( A =/= (/) <-> E. x x e. A ) )
37	2, 36	rexlimddv 2628	1 |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 710   = wceq 1373   E.wex 1515   e. wcel 2176   =/= wne 2376   E.wrex 2485   (/)c0 3460   class class class wbr 4044   suc csuc 4412   omcom 4638   -->wf 5267   -1-1-onto->wf1o 5270   ` cfv 5271   ~~ cen 6825   Fincfn 6827

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 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-13 2178  ax-14 2179  ax-ext 2187  ax-sep 4162  ax-nul 4170  ax-pow 4218  ax-pr 4253  ax-un 4480  ax-iinf 4636

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ne 2377  df-ral 2489  df-rex 2490  df-v 2774  df-sbc 2999  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3461  df-pw 3618  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-int 3886  df-br 4045  df-opab 4106  df-id 4340  df-suc 4418  df-iom 4639  df-xp 4681  df-rel 4682  df-cnv 4683  df-co 4684  df-dm 4685  df-rn 4686  df-res 4687  df-ima 4688  df-iota 5232  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-er 6620  df-en 6828  df-fin 6830

This theorem is referenced by:  findcard2  6986  findcard2s  6987  diffisn  6990  fimax2gtri  6998  elfi2  7074  elfir  7075  fiuni  7080  fifo  7082  4sqlem12  12725