Theorem fidceq 6639

Description: Equality of members of a finite set is decidable. This may be counterintuitive: cannot any two sets be elements of a finite set? Well, to show, for example, that { B , C } is finite would require showing it is equinumerous to 1o or to 2o but to show that you'd need to know B = C or -. B = C, respectively. (Contributed by Jim Kingdon, 5-Sep-2021.)

Assertion

Ref	Expression
fidceq	|- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )

Proof of Theorem fidceq

Dummy variables f x are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		isfi 6532	. . . 4 |- ( A e. Fin <-> E. x e. om A ~~ x )
2	1	biimpi 119	. . 3 |- ( A e. Fin -> E. x e. om A ~~ x )
3	2	3ad2ant1 965	. 2 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> E. x e. om A ~~ x )
4		bren 6518	. . . . 5 |- ( A ~~ x <-> E. f f : A -1-1-onto-> x )
5	4	biimpi 119	. . . 4 |- ( A ~~ x -> E. f f : A -1-1-onto-> x )
6	5	ad2antll 476	. . 3 |- ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) -> E. f f : A -1-1-onto-> x )
7		f1of 5266	. . . . . . . . . 10 |- ( f : A -1-1-onto-> x -> f : A --> x )
8	7	adantl 272	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> f : A --> x )
9		simpll2 984	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> B e. A )
10	8, 9	ffvelrnd 5449	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( f ` B ) e. x )
11		simplrl 503	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> x e. om )
12		elnn 4433	. . . . . . . 8 |- ( ( ( f ` B ) e. x /\ x e. om ) -> ( f ` B ) e. om )
13	10, 11, 12	syl2anc 404	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( f ` B ) e. om )
14		simpll3 985	. . . . . . . . 9 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> C e. A )
15	8, 14	ffvelrnd 5449	. . . . . . . 8 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( f ` C ) e. x )
16		elnn 4433	. . . . . . . 8 |- ( ( ( f ` C ) e. x /\ x e. om ) -> ( f ` C ) e. om )
17	15, 11, 16	syl2anc 404	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( f ` C ) e. om )
18		nndceq 6274	. . . . . . 7 |- ( ( ( f ` B ) e. om /\ ( f ` C ) e. om ) -> DECID ( f ` B ) = ( f ` C ) )
19	13, 17, 18	syl2anc 404	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> DECID ( f ` B ) = ( f ` C ) )
20		exmiddc 783	. . . . . 6 |- (DECID ( f ` B ) = ( f ` C ) -> ( ( f ` B ) = ( f ` C ) \/ -. ( f ` B ) = ( f ` C ) ) )
21	19, 20	syl 14	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( ( f ` B ) = ( f ` C ) \/ -. ( f ` B ) = ( f ` C ) ) )
22		f1of1 5265	. . . . . . . 8 |- ( f : A -1-1-onto-> x -> f : A -1-1-> x )
23	22	adantl 272	. . . . . . 7 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> f : A -1-1-> x )
24		f1veqaeq 5562	. . . . . . 7 |- ( ( f : A -1-1-> x /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
25	23, 9, 14, 24	syl12anc 1173	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
26		fveq2 5318	. . . . . . . 8 |- ( B = C -> ( f ` B ) = ( f ` C ) )
27	26	con3i 598	. . . . . . 7 |- ( -. ( f ` B ) = ( f ` C ) -> -. B = C )
28	27	a1i 9	. . . . . 6 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( -. ( f ` B ) = ( f ` C ) -> -. B = C ) )
29	25, 28	orim12d 736	. . . . 5 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( ( ( f ` B ) = ( f ` C ) \/ -. ( f ` B ) = ( f ` C ) ) -> ( B = C \/ -. B = C ) ) )
30	21, 29	mpd 13	. . . 4 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> ( B = C \/ -. B = C ) )
31		df-dc 782	. . . 4 |- (DECID B = C <-> ( B = C \/ -. B = C ) )
32	30, 31	sylibr 133	. . 3 |- ( ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) /\ f : A -1-1-onto-> x ) -> DECID B = C )
33	6, 32	exlimddv 1827	. 2 |- ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) -> DECID B = C )
34	3, 33	rexlimddv 2494	1 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 665  DECID wdc 781   /\ w3a 925   = wceq 1290   E.wex 1427   e. wcel 1439   E.wrex 2361   class class class wbr 3851   omcom 4418   -->wf 5024   -1-1->wf1 5025   -1-1-onto->wf1o 5027   ` cfv 5028   ~~ cen 6509   Fincfn 6511

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416

This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4457  df-rel 4458  df-cnv 4459  df-co 4460  df-dm 4461  df-rn 4462  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-en 6512  df-fin 6514

This theorem is referenced by:  fidifsnen  6640  fidifsnid  6641  unfiexmid  6682  undiffi  6689  fidcenumlemim  6715