Theorem fidceq 7027

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 6910	. . . 4 |- ( A e. Fin <-> E. x e. om A ~~ x )
2	1	biimpi 120	. . 3 |- ( A e. Fin -> E. x e. om A ~~ x )
3	2	3ad2ant1 1042	. 2 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> E. x e. om A ~~ x )
4		bren 6893	. . . . 5 |- ( A ~~ x <-> E. f f : A -1-1-onto-> x )
5	4	biimpi 120	. . . 4 |- ( A ~~ x -> E. f f : A -1-1-onto-> x )
6	5	ad2antll 491	. . 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 5571	. . . . . . . . . 10 |- ( f : A -1-1-onto-> x -> f : A --> x )
8	7	adantl 277	. . . . . . . . 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 1061	. . . . . . . . 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	ffvelcdmd 5770	. . . . . . . 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 535	. . . . . . . 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 4697	. . . . . . . 8 |- ( ( ( f ` B ) e. x /\ x e. om ) -> ( f ` B ) e. om )
13	10, 11, 12	syl2anc 411	. . . . . . 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 1062	. . . . . . . . 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	ffvelcdmd 5770	. . . . . . . 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 4697	. . . . . . . 8 |- ( ( ( f ` C ) e. x /\ x e. om ) -> ( f ` C ) e. om )
17	15, 11, 16	syl2anc 411	. . . . . . 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 6643	. . . . . . 7 |- ( ( ( f ` B ) e. om /\ ( f ` C ) e. om ) -> DECID ( f ` B ) = ( f ` C ) )
19	13, 17, 18	syl2anc 411	. . . . . 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 841	. . . . . 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 5570	. . . . . . . 8 |- ( f : A -1-1-onto-> x -> f : A -1-1-> x )
23	22	adantl 277	. . . . . . 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 5892	. . . . . . 7 |- ( ( f : A -1-1-> x /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
25	23, 9, 14, 24	syl12anc 1269	. . . . . 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 5626	. . . . . . . 8 |- ( B = C -> ( f ` B ) = ( f ` C ) )
27	26	con3i 635	. . . . . . 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 791	. . . . 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 840	. . . 4 |- (DECID B = C <-> ( B = C \/ -. B = C ) )
32	30, 31	sylibr 134	. . 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 1945	. 2 |- ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) -> DECID B = C )
34	3, 33	rexlimddv 2653	1 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 713  DECID wdc 839   /\ w3a 1002   = wceq 1395   E.wex 1538   e. wcel 2200   E.wrex 2509   class class class wbr 4082   omcom 4681   -->wf 5313   -1-1->wf1 5314   -1-1-onto->wf1o 5316   ` cfv 5317   ~~ cen 6883   Fincfn 6885

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523  ax-setind 4628  ax-iinf 4679

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-int 3923  df-br 4083  df-opab 4145  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-iom 4682  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-en 6886  df-fin 6888

This theorem is referenced by:  fidifsnen  7028  fidifsnid  7029  pw1fin  7068  unfiexmid  7076  undiffi  7083  fidcenumlemim  7115