Theorem fin0or 6944

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 6817	. . 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 4637	. . . 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 4056	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
8		en0 6851	. . . . . 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 6839	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
14		bren 6803	. . . . . . . 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 5501	. . . . . . . . . 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 4448	. . . . . . . . . . 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 2273	. . . . . . . . 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 5695	. . . . . . . 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 2776	. . . . . . . 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 1910	. . . . . 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 2611	. . . 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 2616	1 |- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2164   E.wrex 2473   (/)c0 3447   class class class wbr 4030   suc csuc 4397   omcom 4623   -->wf 5251   -1-1-onto->wf1o 5254   ` cfv 5255   ~~ cen 6794   Fincfn 6796

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-nul 4156  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-iinf 4621

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-sbc 2987  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3448  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-int 3872  df-br 4031  df-opab 4092  df-id 4325  df-suc 4403  df-iom 4624  df-xp 4666  df-rel 4667  df-cnv 4668  df-co 4669  df-dm 4670  df-rn 4671  df-res 4672  df-ima 4673  df-iota 5216  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-er 6589  df-en 6797  df-fin 6799

This theorem is referenced by:  xpfi  6988  fival  7031  fiubm  10902  fsumcllem  11545  fprodcllem  11752  gsumwsubmcl  13071  gsumwmhm  13073