Theorem fin0or 6947

Description: A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)

Assertion

Ref	Expression
fin0or	|- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )

Distinct variable group:   x, A

Proof of Theorem fin0or

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

Step	Hyp	Ref	Expression
1		isfi 6820	. . 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		nn0suc 4640	. . . 4 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
4	3	ad2antrl 490	. . 3 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) \/ E. m e. om n = suc m ) )
5		simplrr 536	. . . . . . 7 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
6		simpr 110	. . . . . . 7 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> n = (/) )
7	5, 6	breqtrd 4059	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
8		en0 6854	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
9	7, 8	sylib 122	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
10	9	ex 115	. . . 4 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) -> A = (/) ) )
11		simplrr 536	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> A ~~ n )
12	11	adantr 276	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ n )
13	12	ensymd 6842	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
14		bren 6806	. . . . . . . 8 |- ( n ~~ A <-> E. f f : n -1-1-onto-> A )
15	13, 14	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 )
16		f1of 5504	. . . . . . . . . 10 |- ( f : n -1-1-onto-> A -> f : n --> A )
17	16	adantl 277	. . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> f : n --> A )
18		sucidg 4451	. . . . . . . . . . 11 |- ( m e. om -> m e. suc m )
19	18	ad3antlr 493	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> m e. suc m )
20		simplr 528	. . . . . . . . . 10 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> n = suc m )
21	19, 20	eleqtrrd 2276	. . . . . . . . 9 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> m e. n )
22	17, 21	ffvelcdmd 5698	. . . . . . . 8 |- ( ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) /\ f : n -1-1-onto-> A ) -> ( f ` m ) e. A )
23		elex2 2779	. . . . . . . 8 |- ( ( f ` m ) e. A -> E. x x e. A )
24	22, 23	syl 14	. . . . . . 7 |- ( ( ( ( ( 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 )
25	15, 24	exlimddv 1913	. . . . . 6 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> E. x x e. A )
26	25	ex 115	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> ( n = suc m -> E. x x e. A ) )
27	26	rexlimdva 2614	. . . 4 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( E. m e. om n = suc m -> E. x x e. A ) )
28	10, 27	orim12d 787	. . 3 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( ( n = (/) \/ E. m e. om n = suc m ) -> ( A = (/) \/ E. x x e. A ) ) )
29	4, 28	mpd 13	. 2 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( A = (/) \/ E. x x e. A ) )
30	2, 29	rexlimddv 2619	1 |- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1506   e. wcel 2167   E.wrex 2476   (/)c0 3450   class class class wbr 4033   suc csuc 4400   omcom 4626   -->wf 5254   -1-1-onto->wf1o 5257   ` cfv 5258   ~~ cen 6797   Fincfn 6799

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 4151  ax-nul 4159  ax-pow 4207  ax-pr 4242  ax-un 4468  ax-iinf 4624

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-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 3451  df-pw 3607  df-sn 3628  df-pr 3629  df-op 3631  df-uni 3840  df-int 3875  df-br 4034  df-opab 4095  df-id 4328  df-suc 4406  df-iom 4627  df-xp 4669  df-rel 4670  df-cnv 4671  df-co 4672  df-dm 4673  df-rn 4674  df-res 4675  df-ima 4676  df-iota 5219  df-fun 5260  df-fn 5261  df-f 5262  df-f1 5263  df-fo 5264  df-f1o 5265  df-fv 5266  df-er 6592  df-en 6800  df-fin 6802

This theorem is referenced by:  xpfi  6993  fival  7036  fiubm  10920  fsumcllem  11564  fprodcllem  11771  gsumwsubmcl  13128  gsumwmhm  13130