Theorem diffisn 6963

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 6829	. . . 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 2779	. . . . . . . . 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 6955	. . . . . . . . 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 2388	. . . . 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 6863	. . . . . . . . 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 6852	. . . . . . 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 6863	. . . . . 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 4641	. . . . . 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 6942	. . . . 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 4038	. . . . . . 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 6949	. . . . 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 6944	. . . 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 2619	. 2 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { B } ) e. Fin )
37	3, 36	rexlimddv 2619	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 1506   e. wcel 2167   =/= wne 2367   E.wrex 2476   \ cdif 3154   (/)c0 3451   {csn 3623   class class class wbr 4034   suc csuc 4401   omcom 4627   ~~ cen 6806   Fincfn 6808

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-13 2169  ax-14 2170  ax-ext 2178  ax-coll 4149  ax-sep 4152  ax-nul 4160  ax-pow 4208  ax-pr 4243  ax-un 4469  ax-setind 4574  ax-iinf 4625

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 1475  df-sb 1777  df-eu 2048  df-mo 2049  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ne 2368  df-ral 2480  df-rex 2481  df-reu 2482  df-rab 2484  df-v 2765  df-sbc 2990  df-csb 3085  df-dif 3159  df-un 3161  df-in 3163  df-ss 3170  df-nul 3452  df-if 3563  df-pw 3608  df-sn 3629  df-pr 3630  df-op 3632  df-uni 3841  df-int 3876  df-iun 3919  df-br 4035  df-opab 4096  df-mpt 4097  df-tr 4133  df-id 4329  df-iord 4402  df-on 4404  df-suc 4407  df-iom 4628  df-xp 4670  df-rel 4671  df-cnv 4672  df-co 4673  df-dm 4674  df-rn 4675  df-res 4676  df-ima 4677  df-iota 5220  df-fun 5261  df-fn 5262  df-f 5263  df-f1 5264  df-fo 5265  df-f1o 5266  df-fv 5267  df-er 6601  df-en 6809  df-fin 6811

This theorem is referenced by:  diffifi  6964  zfz1isolemsplit  10947  zfz1isolem1  10949  fsumdifsnconst  11637  fprodeq0g  11820