Theorem diffisn 6906

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 6774	. . . 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 2765	. . . . . . . . 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 6898	. . . . . . . . 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 2378	. . . . 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 6808	. . . . . . . . 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 6797	. . . . . . 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 6808	. . . . . 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 4615	. . . . . 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 1359	. . 3 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> E. m e. om n = suc m )
24		nnfi 6885	. . . . 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 4019	. . . . . . 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 6892	. . . . 5 |- ( ( m e. om /\ A ~~ suc m /\ B e. A ) -> ( A \ { B } ) ~~ m )
33	26, 30, 31, 32	syl3anc 1248	. . . 4 |- ( ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) /\ ( m e. om /\ n = suc m ) ) -> ( A \ { B } ) ~~ m )
34		enfii 6887	. . . 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 2609	. 2 |- ( ( ( A e. Fin /\ B e. A ) /\ ( n e. om /\ A ~~ n ) ) -> ( A \ { B } ) e. Fin )
37	3, 36	rexlimddv 2609	1 |- ( ( A e. Fin /\ B e. A ) -> ( A \ { B } ) e. Fin )

Syntax hints:   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 709   = wceq 1363   E.wex 1502   e. wcel 2158   =/= wne 2357   E.wrex 2466   \ cdif 3138   (/)c0 3434   {csn 3604   class class class wbr 4015   suc csuc 4377   omcom 4601   ~~ cen 6751   Fincfn 6753

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 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-coll 4130  ax-sep 4133  ax-nul 4141  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-iinf 4599

This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 980  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-ral 2470  df-rex 2471  df-reu 2472  df-rab 2474  df-v 2751  df-sbc 2975  df-csb 3070  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-if 3547  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-int 3857  df-iun 3900  df-br 4016  df-opab 4077  df-mpt 4078  df-tr 4114  df-id 4305  df-iord 4378  df-on 4380  df-suc 4383  df-iom 4602  df-xp 4644  df-rel 4645  df-cnv 4646  df-co 4647  df-dm 4648  df-rn 4649  df-res 4650  df-ima 4651  df-iota 5190  df-fun 5230  df-fn 5231  df-f 5232  df-f1 5233  df-fo 5234  df-f1o 5235  df-fv 5236  df-er 6548  df-en 6754  df-fin 6756

This theorem is referenced by:  diffifi  6907  zfz1isolemsplit  10831  zfz1isolem1  10833  fsumdifsnconst  11476  fprodeq0g  11659