Theorem eqsndc 7094

Description: Decidability of equality between a finite subset of a set with decidable equality, and a singleton whose element is an element of the larger set. (Contributed by Jim Kingdon, 15-Feb-2026.)

Hypotheses

Ref	Expression
elssdc.b	|- ( ph -> A. x e. B A. y e. B DECID x = y )
elssdc.x	|- ( ph -> X e. B )
elssdc.ss	|- ( ph -> A C_ B )
elssdc.a	|- ( ph -> A e. Fin )

Assertion

Ref	Expression
eqsndc	|- ( ph -> DECID A = { X } )

Distinct variable groups:   x, B, y   x, X, y

Allowed substitution hints:   ph( x, y)   A( x, y)

Proof of Theorem eqsndc

Dummy variable u is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . 5 |- ( ( ph /\ A ~~ { X } ) -> A ~~ { X } )
2		elssdc.x	. . . . . . 7 |- ( ph -> X e. B )
3		ensn1g 6970	. . . . . . 7 |- ( X e. B -> { X } ~~ 1o )
4	2, 3	syl 14	. . . . . 6 |- ( ph -> { X } ~~ 1o )
5	4	adantr 276	. . . . 5 |- ( ( ph /\ A ~~ { X } ) -> { X } ~~ 1o )
6		entr 6957	. . . . 5 |- ( ( A ~~ { X } /\ { X } ~~ 1o ) -> A ~~ 1o )
7	1, 5, 6	syl2anc 411	. . . 4 |- ( ( ph /\ A ~~ { X } ) -> A ~~ 1o )
8		en1 6972	. . . 4 |- ( A ~~ 1o <-> E. u A = { u } )
9	7, 8	sylib 122	. . 3 |- ( ( ph /\ A ~~ { X } ) -> E. u A = { u } )
10		elssdc.ss	. . . . . . 7 |- ( ph -> A C_ B )
11	10	ad2antrr 488	. . . . . 6 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> A C_ B )
12		vsnid 3701	. . . . . . . 8 |- u e. { u }
13		eleq2 2295	. . . . . . . 8 |- ( A = { u } -> ( u e. A <-> u e. { u } ) )
14	12, 13	mpbiri 168	. . . . . . 7 |- ( A = { u } -> u e. A )
15	14	adantl 277	. . . . . 6 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> u e. A )
16	11, 15	sseldd 3228	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> u e. B )
17	2	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> X e. B )
18		elssdc.b	. . . . . 6 |- ( ph -> A. x e. B A. y e. B DECID x = y )
19	18	ad2antrr 488	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> A. x e. B A. y e. B DECID x = y )
20		eqeq1 2238	. . . . . . 7 |- ( x = u -> ( x = y <-> u = y ) )
21	20	dcbid 845	. . . . . 6 |- ( x = u -> (DECID x = y <-> DECID u = y ) )
22		eqeq2 2241	. . . . . . 7 |- ( y = X -> ( u = y <-> u = X ) )
23	22	dcbid 845	. . . . . 6 |- ( y = X -> (DECID u = y <-> DECID u = X ) )
24	21, 23	rspc2va 2924	. . . . 5 |- ( ( ( u e. B /\ X e. B ) /\ A. x e. B A. y e. B DECID x = y ) -> DECID u = X )
25	16, 17, 19, 24	syl21anc 1272	. . . 4 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> DECID u = X )
26		eqeq1 2238	. . . . . . 7 |- ( A = { u } -> ( A = { X } <-> { u } = { X } ) )
27	26	adantl 277	. . . . . 6 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> ( A = { X } <-> { u } = { X } ) )
28		sneqbg 3846	. . . . . . 7 |- ( u e. _V -> ( { u } = { X } <-> u = X ) )
29	28	elv 2806	. . . . . 6 |- ( { u } = { X } <-> u = X )
30	27, 29	bitrdi 196	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> ( A = { X } <-> u = X ) )
31	30	dcbid 845	. . . 4 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> (DECID A = { X } <-> DECID u = X ) )
32	25, 31	mpbird 167	. . 3 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> DECID A = { X } )
33	9, 32	exlimddv 1947	. 2 |- ( ( ph /\ A ~~ { X } ) -> DECID A = { X } )
34		elssdc.a	. . . . . 6 |- ( ph -> A e. Fin )
35		eqeng 6938	. . . . . 6 |- ( A e. Fin -> ( A = { X } -> A ~~ { X } ) )
36	34, 35	syl 14	. . . . 5 |- ( ph -> ( A = { X } -> A ~~ { X } ) )
37	36	con3dimp 640	. . . 4 |- ( ( ph /\ -. A ~~ { X } ) -> -. A = { X } )
38	37	olcd 741	. . 3 |- ( ( ph /\ -. A ~~ { X } ) -> ( A = { X } \/ -. A = { X } ) )
39		df-dc 842	. . 3 |- (DECID A = { X } <-> ( A = { X } \/ -. A = { X } ) )
40	38, 39	sylibr 134	. 2 |- ( ( ph /\ -. A ~~ { X } ) -> DECID A = { X } )
41		snfig 6988	. . . . 5 |- ( X e. B -> { X } e. Fin )
42	2, 41	syl 14	. . . 4 |- ( ph -> { X } e. Fin )
43		fidcen 7087	. . . 4 |- ( ( A e. Fin /\ { X } e. Fin ) -> DECID A ~~ { X } )
44	34, 42, 43	syl2anc 411	. . 3 |- ( ph -> DECID A ~~ { X } )
45		exmiddc 843	. . 3 |- (DECID A ~~ { X } -> ( A ~~ { X } \/ -. A ~~ { X } ) )
46	44, 45	syl 14	. 2 |- ( ph -> ( A ~~ { X } \/ -. A ~~ { X } ) )
47	33, 40, 46	mpjaodan 805	1 |- ( ph -> DECID A = { X } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 715  DECID wdc 841   = wceq 1397   E.wex 1540   e. wcel 2202   A.wral 2510   _Vcvv 2802   C_ wss 3200   {csn 3669   class class class wbr 4088   1oc1o 6574   ~~ 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-fal 1403  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-reu 2517  df-rab 2519  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-res 4737  df-ima 4738  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-1o 6581  df-er 6701  df-en 6909  df-fin 6911

This theorem is referenced by:  vtxlpfi  16140