Theorem diffisn 7163

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 7013	. . . 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 2832	. . . . . . . . 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 7155	. . . . . . . . 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 2435	. . . . 5 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> -. A = (/) )
11		simplrr 538	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
12		en0 7048	. . . . . . . . 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 7037	. . . . . . 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 7048	. . . . . 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 671	. . . 4 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> -. n = (/) )
20		nn0suc 4731	. . . . . 6 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
21	20	orcomd 737	. . . . 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 1386	. . 3 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om n = suc m )
24		nnfi 7140	. . . . 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 531	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> m e. om )
27		simplrr 538	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> A ~~ n )
28		breq2 4118	. . . . . . 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 536	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> B e. A )
32		dif1en 7149	. . . . 5 |- ( ( m e. om /\ A ~~ suc m /\ B e. A ) -> ( A \ { B } ) ~~ m )
33	26, 30, 31, 32	syl3anc 1274	. . . 4 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> ( A \ { B } ) ~~ m )
34		enfii 7142	. . . 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 2667	. 2 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { B } ) e. Fin )
37	3, 36	rexlimddv 2667	1 |- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716   = wceq 1398   E.wex 1541   e. wcel 2205   =/= wne 2414   E.wrex 2523   \ cdif 3211   (/)c0 3512   {csn 3694   class class class wbr 4114   suc csuc 4491   omcom 4717   ~~ cen 6986   Fincfn 6988

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-coll 4230  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-reu 2529  df-rab 2531  df-v 2817  df-sbc 3046  df-csb 3142  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-if 3625  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-iun 3998  df-br 4115  df-opab 4177  df-mpt 4178  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-ima 4767  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-er 6780  df-en 6989  df-fin 6991

This theorem is referenced by:  diffifi  7164  hashfibclem  11231  zfz1isolemsplit  11235  zfz1isolem1  11237  fsumdifsnconst  12166  fprodeq0g  12349