Theorem fin0or 6864

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 6739	. . 3 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 119	. 2 |- ( A e. Fin -> E. n e. om A ~~ n )
3		nn0suc 4588	. . . 4 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
4	3	ad2antrl 487	. . 3 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) \/ E. m e. om n = suc m ) )
5		simplrr 531	. . . . . . 7 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
6		simpr 109	. . . . . . 7 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> n = (/) )
7	5, 6	breqtrd 4015	. . . . . 6 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
8		en0 6773	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
9	7, 8	sylib 121	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
10	9	ex 114	. . . 4 |- ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) -> ( n = (/) -> A = (/) ) )
11		simplrr 531	. . . . . . . . . 10 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> A ~~ n )
12	11	adantr 274	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> A ~~ n )
13	12	ensymd 6761	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> n ~~ A )
14		bren 6725	. . . . . . . 8 |- ( n ~~ A <-> E. f f : n -1-1-onto-> A )
15	13, 14	sylib 121	. . . . . . 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 5442	. . . . . . . . . 10 |- ( f : n -1-1-onto-> A -> f : n --> A )
17	16	adantl 275	. . . . . . . . 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 4401	. . . . . . . . . . 11 |- ( m e. om -> m e. suc m )
19	18	ad3antlr 490	. . . . . . . . . 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 525	. . . . . . . . . 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 2250	. . . . . . . . 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	ffvelrnd 5632	. . . . . . . 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 2746	. . . . . . . 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 1891	. . . . . 6 |- ( ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) /\ n = suc m ) -> E. x x e. A )
26	25	ex 114	. . . . 5 |- ( ( ( A e. Fin /\ ( n e. om /\ A ~~ n ) ) /\ m e. om ) -> ( n = suc m -> E. x x e. A ) )
27	26	rexlimdva 2587	. . . 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 781	. . 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 2592	1 |- ( A e. Fin -> ( A = (/) \/ E. x x e. A ) )

Syntax hints:   -> wi 4   /\ wa 103   \/ wo 703   = wceq 1348   E.wex 1485   e. wcel 2141   E.wrex 2449   (/)c0 3414   class class class wbr 3989   suc csuc 4350   omcom 4574   -->wf 5194   -1-1-onto->wf1o 5197   ` cfv 5198   ~~ cen 6716   Fincfn 6718

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-id 4278  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-er 6513  df-en 6719  df-fin 6721

This theorem is referenced by:  xpfi  6907  fival  6947  fiubm  10763  fsumcllem  11362  fprodcllem  11569