Theorem fin0or 7068

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 6929	. . 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 4700	. . . 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 4112	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
8		en0 6964	. . . . . 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 6952	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
14		bren 6912	. . . . . . . 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 5580	. . . . . . . . . 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 4511	. . . . . . . . . . 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 2309	. . . . . . . . 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 5779	. . . . . . . 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 2817	. . . . . . . 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 1945	. . . . . 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 2648	. . . 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 791	. . 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 2653	1 |- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )

Syntax hints:   -> wi 4   /\ wa 104   \/ wo 713   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   (/)c0 3492   class class class wbr 4086   suc csuc 4460   omcom 4686   -->wf 5320   -1-1-onto->wf1o 5323   ` cfv 5324   ~~ cen 6902   Fincfn 6904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-id 4388  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-er 6697  df-en 6905  df-fin 6907

This theorem is referenced by:  xpfi  7117  fival  7160  fiubm  11082  lswex  11155  fsumcllem  11950  fprodcllem  12157  gsumwsubmcl  13569  gsumwmhm  13571