Theorem diffisn 6350

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 6241	. . . 4 |- ( A e. Fin <-> E. n e. om A ~~ n )
2	1	biimpi 113	. . 3 |- ( A e. Fin -> E. n e. om A ~~ n )
3	2	adantr 261	. 2 |- ( ( A e. Fin /\ B e. A ) -> E. n e. om A ~~ n )
4		elex2 2570	. . . . . . . . 9 |- ( B e. A -> E. x x e. A )
5	4	adantl 262	. . . . . . . 8 |- ( ( A e. Fin /\ B e. A ) -> E. x x e. A )
6		fin0 6342	. . . . . . . . 9 |- ( A e. Fin -> ( A =/= (/) <-> E. x x e. A ) )
7	6	adantr 261	. . . . . . . 8 |- ( ( A e. Fin /\ B e. A ) -> ( A =/= (/) <-> E. x x e. A ) )
8	5, 7	mpbird 156	. . . . . . 7 |- ( ( A e. Fin /\ B e. A ) -> A =/= (/) )
9	8	adantr 261	. . . . . 6 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> A =/= (/) )
10	9	neneqd 2226	. . . . 5 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> -. A = (/) )
11		simplrr 488	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ n )
12		en0 6275	. . . . . . . . 9 |- ( n ~~ (/) <-> n = (/) )
13	12	biimpri 124	. . . . . . . 8 |- ( n = (/) -> n ~~ (/) )
14	13	adantl 262	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> n ~~ (/) )
15		entr 6264	. . . . . . 7 |- ( ( A ~~ n /\ n ~~ (/) ) -> A ~~ (/) )
16	11, 14, 15	syl2anc 391	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A ~~ (/) )
17		en0 6275	. . . . . 6 |- ( A ~~ (/) <-> A = (/) )
18	16, 17	sylib 127	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ n = (/) ) -> A = (/) )
19	10, 18	mtand 591	. . . 4 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> -. n = (/) )
20		nn0suc 4327	. . . . . 6 |- ( n e. om -> ( n = (/) \/ E. m e. om n = suc m ) )
21	20	orcomd 648	. . . . 5 |- ( n e. om -> ( E. m e. om n = suc m \/ n = (/) ) )
22	21	ad2antrl 459	. . . 4 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> ( E. m e. om n = suc m \/ n = (/) ) )
23	19, 22	ecased 1239	. . 3 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om n = suc m )
24		nnfi 6333	. . . . 5 |- ( m e. om -> m e. Fin )
25	24	ad2antrl 459	. . . 4 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> m e. Fin )
26		simprl 483	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> m e. om )
27		simplrr 488	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> A ~~ n )
28		breq2 3768	. . . . . . 7 |- ( n = suc m -> ( A ~~ n <-> A ~~ suc m ) )
29	28	ad2antll 460	. . . . . 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 135	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> A ~~ suc m )
31		simpllr 486	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> B e. A )
32		dif1en 6337	. . . . 5 |- ( ( m e. om /\ A ~~ suc m /\ B e. A ) -> ( A \ { B } ) ~~ m )
33	26, 30, 31, 32	syl3anc 1135	. . . 4 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> ( A \ { B } ) ~~ m )
34		enfii 6335	. . . 4 |- ( ( m e. Fin /\ ( A \ { B } ) ~~ m ) -> ( A \ { B } ) e. Fin )
35	25, 33, 34	syl2anc 391	. . 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 2437	. 2 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { B } ) e. Fin )
37	3, 36	rexlimddv 2437	1 |- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 97   <-> wb 98   \/ wo 629   = wceq 1243   E.wex 1381   e. wcel 1393   =/= wne 2204   E.wrex 2307   \ cdif 2914   (/)c0 3224   {csn 3375   class class class wbr 3764   suc csuc 4102   omcom 4313   ~~ cen 6219   Fincfn 6221

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-iinf 4311

This theorem depends on definitions:  df-bi 110  df-dc 743  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-if 3332  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-tr 3855  df-id 4030  df-iord 4103  df-on 4105  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-er 6106  df-en 6222  df-fin 6224

This theorem is referenced by:  diffifi  6351