Theorem fidceq 7137

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 7013	. . . 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 1045	. 2 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> E. x e. om A ~~ x )
4		bren 6996	. . . . 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 5619	. . . . . . . . . 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 1064	. . . . . . . . 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 5818	. . . . . . . 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 537	. . . . . . . 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 4733	. . . . . . . 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 1065	. . . . . . . . 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 5818	. . . . . . . 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 4733	. . . . . . . 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 6745	. . . . . . 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 844	. . . . . 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 5618	. . . . . . . 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 5948	. . . . . . 7 |- ( ( f : A -1-1-> x /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
25	23, 9, 14, 24	syl12anc 1272	. . . . . 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 5675	. . . . . . . 8 |- ( B = C -> ( f ` B ) = ( f ` C ) )
27	26	con3i 637	. . . . . . 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 794	. . . . 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 843	. . . 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 1950	. 2 |- ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) -> DECID B = C )
34	3, 33	rexlimddv 2667	1 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 716  DECID wdc 842   /\ w3a 1005   = wceq 1398   E.wex 1541   e. wcel 2205   E.wrex 2523   class class class wbr 4114   omcom 4717   -->wf 5353   -1-1->wf1 5354   -1-1-onto->wf1o 5356   ` cfv 5357   ~~ cen 6986   Fincfn 6988

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-nul 4241  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-iinf 4715

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-int 3955  df-br 4115  df-opab 4177  df-tr 4214  df-id 4419  df-iord 4492  df-on 4494  df-suc 4497  df-iom 4718  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-f1 5362  df-fo 5363  df-f1o 5364  df-fv 5365  df-en 6989  df-fin 6991

This theorem is referenced by:  fidifsnen  7138  fidifsnid  7139  pw1fin  7183  unfiexmid  7191  undiffi  7198  fissfi  7229  fidcenumlemim  7235  ballotfilem2  13172  vtxedgfi  16410  vtxlpfi  16411  eupth2lem3lem3fi  16591  eupth2lem3lem4fi  16594  eupth2lem3lem7fi  16595  eupth2lemsfi  16599  eulerpathprum  16601