Theorem fidceq 7055

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 6933	. . . 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 1044	. 2 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> E. x e. om A ~~ x )
4		bren 6916	. . . . 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 5583	. . . . . . . . . 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 1063	. . . . . . . . 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 5783	. . . . . . . 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 4704	. . . . . . . 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 1064	. . . . . . . . 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 5783	. . . . . . . 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 4704	. . . . . . . 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 6666	. . . . . . 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 843	. . . . . 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 5582	. . . . . . . 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 5909	. . . . . . 7 |- ( ( f : A -1-1-> x /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
25	23, 9, 14, 24	syl12anc 1271	. . . . . 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 5639	. . . . . . . 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 793	. . . . 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 842	. . . 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 1947	. 2 |- ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) -> DECID B = C )
34	3, 33	rexlimddv 2655	1 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 715  DECID wdc 841   /\ w3a 1004   = wceq 1397   E.wex 1540   e. wcel 2202   E.wrex 2511   class class class wbr 4088   omcom 4688   -->wf 5322   -1-1->wf1 5323   -1-1-onto->wf1o 5325   ` cfv 5326   ~~ cen 6906   Fincfn 6908

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-en 6909  df-fin 6911

This theorem is referenced by:  fidifsnen  7056  fidifsnid  7057  pw1fin  7101  unfiexmid  7109  undiffi  7116  fidcenumlemim  7150  vtxedgfi  16139  vtxlpfi  16140