Theorem fidceq 6863

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 6755	. . . 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 1018	. 2 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> E. x e. om A ~~ x )
4		bren 6741	. . . . 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 5457	. . . . . . . . . 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 1037	. . . . . . . . 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 5648	. . . . . . . 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 4602	. . . . . . . 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 1038	. . . . . . . . 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 5648	. . . . . . . 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 4602	. . . . . . . 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 6494	. . . . . . 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 836	. . . . . 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 5456	. . . . . . . 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 5764	. . . . . . 7 |- ( ( f : A -1-1-> x /\ ( B e. A /\ C e. A ) ) -> ( ( f ` B ) = ( f ` C ) -> B = C ) )
25	23, 9, 14, 24	syl12anc 1236	. . . . . 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 5511	. . . . . . . 8 |- ( B = C -> ( f ` B ) = ( f ` C ) )
27	26	con3i 632	. . . . . . 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 786	. . . . 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 835	. . . 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 1898	. 2 |- ( ( ( A e. Fin /\ B e. A /\ C e. A ) /\ ( x e. om /\ A ~~ x ) ) -> DECID B = C )
34	3, 33	rexlimddv 2599	1 |- ( ( A e. Fin /\ B e. A /\ C e. A ) -> DECID B = C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 708  DECID wdc 834   /\ w3a 978   = wceq 1353   E.wex 1492   e. wcel 2148   E.wrex 2456   class class class wbr 4000   omcom 4586   -->wf 5208   -1-1->wf1 5209   -1-1-onto->wf1o 5211   ` cfv 5212   ~~ cen 6732   Fincfn 6734

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584

This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-v 2739  df-sbc 2963  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-br 4001  df-opab 4062  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-en 6735  df-fin 6737

This theorem is referenced by:  fidifsnen  6864  fidifsnid  6865  pw1fin  6904  unfiexmid  6911  undiffi  6918  fidcenumlemim  6945