Theorem diffisn 6936

Description: Subtracting a singleton from a finite set produces a finite set. (Contributed by Jim Kingdon, 11-Sep-2021.)

Assertion

Ref	Expression
diffisn	|- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )

Proof of Theorem diffisn

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

Step	Hyp	Ref	Expression
1		isfi 6802	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 120	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	adantr 276	. 2 |- ( ( A e. Fin /\ B e. A ) -> E. n e. om A ~~ n )
4		elex2 2772	. . . . . . . . 9 |- ( B e. A -> E. x x e. A )
5	4	adantl 277	. . . . . . . 8 |- ( ( A e. Fin /\ B e. A ) -> E. x x e. A )
6		fin0 6928	. . . . . . . . 9 |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )
7	6	adantr 276	. . . . . . . 8 |- ( ( A e. Fin /\ B e. A ) -> ( A =/= (/) <-> E. x x e. A ) )
8	5, 7	mpbird 167	. . . . . . 7 |- ( ( A e. Fin /\ B e. A ) -> A =/= (/) )
9	8	adantr 276	. . . . . 6 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> A =/= (/) )
10	9	neneqd 2381	. . . . 5 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> -. A = (/) )
11		simplrr 536	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
12		en0 6836	. . . . . . . . 9 |- ( n ~~ (/) <-> n = (/) )
13	12	biimpri 133	. . . . . . . 8 |- ( n = (/) -> n ~~ (/) )
14	13	adantl 277	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> n ~~ (/) )
15		entr 6825	. . . . . . 7 |- ( ( A ~~ n /\ n ~~ (/) ) -> A ~~ (/) )
16	11, 14, 15	syl2anc 411	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
17		en0 6836	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
18	16, 17	sylib 122	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
19	10, 18	mtand 666	. . . 4 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> -. n = (/) )
20		nn0suc 4628	. . . . . 6 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
21	20	orcomd 730	. . . . 5 |- ( n e. om -> ( E. m e. om n = suc m \/ n = (/) ) )
22	21	ad2antrl 490	. . . 4 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> ( E. m e. om n = suc m \/ n = (/) ) )
23	19, 22	ecased 1360	. . 3 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om n = suc m )
24		nnfi 6915	. . . . 5 |- ( m e. om -> m e. Fin )
25	24	ad2antrl 490	. . . 4 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> m e. Fin )
26		simprl 529	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> m e. om )
27		simplrr 536	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> A ~~ n )
28		breq2 4029	. . . . . . 7 |- ( n = suc m -> ( A ~~ n <-> A ~~ suc m ) )
29	28	ad2antll 491	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> ( A ~~ n <-> A ~~ suc m ) )
30	27, 29	mpbid 147	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> A ~~ suc m )
31		simpllr 534	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> B e. A )
32		dif1en 6922	. . . . 5 |- ( ( m e. om /\ A ~~ suc m /\ B e. A ) -> ( A \ { B } ) ~~ m )
33	26, 30, 31, 32	syl3anc 1249	. . . 4 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> ( A \ { B } ) ~~ m )
34		enfii 6917	. . . 4 |- ( ( m e. Fin /\ ( A \ { B } ) ~~ m ) -> ( A \ { B } ) e. Fin )
35	25, 33, 34	syl2anc 411	. . 3 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> ( A \ { B } ) e. Fin )
36	23, 35	rexlimddv 2612	. 2 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { B } ) e. Fin )
37	3, 36	rexlimddv 2612	1 |- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1364   E.wex 1503   e. wcel 2160   =/= wne 2360   E.wrex 2469   \ cdif 3146   (/)c0 3442   {csn 3614   class class class wbr 4025   suc csuc 4390   omcom 4614   ~~ cen 6779   Fincfn 6781

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 2162  ax-14 2163  ax-ext 2171  ax-coll 4140  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2758  df-sbc 2982  df-csb 3077  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-if 3554  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-iun 3910  df-br 4026  df-opab 4087  df-mpt 4088  df-tr 4124  df-id 4318  df-iord 4391  df-on 4393  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-er 6574  df-en 6782  df-fin 6784

This theorem is referenced by:  diffifi  6937  zfz1isolemsplit  10883  zfz1isolem1  10885  fsumdifsnconst  11572  fprodeq0g  11755