Theorem eqsndc 7210

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 7084	. . . . . . 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 7071	. . . . 5 |- ( ( A ~~ { X } /\ { X } ~~ 1o ) -> A ~~ 1o )
7	1, 5, 6	syl2anc 415	. . . 4 |- ( ( ph /\ A ~~ { X } ) -> A ~~ 1o )
8		en1 7086	. . . 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 492	. . . . . 6 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> A C_ B )
12		vsnid 3741	. . . . . . . 8 |- u e. { u }
13		eleq2 2302	. . . . . . . 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 3249	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> u e. B )
17	2	ad2antrr 492	. . . . 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 492	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> A. x e. B A. y e. B DECID x = y )
20		eqeq1 2245	. . . . . . 7 |- ( x = u -> ( x = y <-> u = y ) )
21	20	dcbid 850	. . . . . 6 |- ( x = u -> (DECID x = y <-> DECID u = y ) )
22		eqeq2 2248	. . . . . . 7 |- ( y = X -> ( u = y <-> u = X ) )
23	22	dcbid 850	. . . . . 6 |- ( y = X -> (DECID u = y <-> DECID u = X ) )
24	21, 23	rspc2va 2944	. . . . 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 1277	. . . 4 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> DECID u = X )
26		eqeq1 2245	. . . . . . 7 |- ( A = { u } -> ( A = { X } <-> { u } = { X } ) )
27	26	adantl 277	. . . . . 6 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> ( A = { X } <-> { u } = { X } ) )
28		sneqbg 3888	. . . . . . 7 |- ( u e. _V -> ( { u } = { X } <-> u = X ) )
29	28	elv 2825	. . . . . 6 |- ( { u } = { X } <-> u = X )
30	27, 29	bitrdi 196	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> ( A = { X } <-> u = X ) )
31	30	dcbid 850	. . . 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 1954	. 2 |- ( ( ph /\ A ~~ { X } ) -> DECID A = { X } )
34		elssdc.a	. . . . . 6 |- ( ph -> A e. Fin )
35		eqeng 7052	. . . . . 6 |- ( A e. Fin -> ( A = { X } -> A ~~ { X } ) )
36	34, 35	syl 14	. . . . 5 |- ( ph -> ( A = { X } -> A ~~ { X } ) )
37	36	con3dimp 644	. . . 4 |- ( ( ph /\ -. A ~~ { X } ) -> -. A = { X } )
38	37	olcd 746	. . 3 |- ( ( ph /\ -. A ~~ { X } ) -> ( A = { X } \/ -. A = { X } ) )
39		df-dc 847	. . 3 |- (DECID A = { X } <-> ( A = { X } \/ -. A = { X } ) )
40	38, 39	sylibr 134	. 2 |- ( ( ph /\ -. A ~~ { X } ) -> DECID A = { X } )
41		snfig 7103	. . . . 5 |- ( X e. B -> { X } e. Fin )
42	2, 41	syl 14	. . . 4 |- ( ph -> { X } e. Fin )
43		fidcen 7203	. . . 4 |- ( ( A e. Fin /\ { X } e. Fin ) -> DECID A ~~ { X } )
44	34, 42, 43	syl2anc 415	. . 3 |- ( ph -> DECID A ~~ { X } )
45		exmiddc 848	. . 3 |- (DECID A ~~ { X } -> ( A ~~ { X } \/ -. A ~~ { X } ) )
46	44, 45	syl 14	. 2 |- ( ph -> ( A ~~ { X } \/ -. A ~~ { X } ) )
47	33, 40, 46	mpjaodan 810	1 |- ( ph -> DECID A = { X } )

This proof depends on syntax axioms:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 720  DECID wdc 846   = wceq 1402   E.wex 1545   e. wcel 2209   A.wral 2528   _Vcvv 2821   C_ wss 3220   {csn 3709   class class class wbr 4130   1oc1o 6680   ~~ cen 7020   Fincfn 7022

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-nul 4259  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684  ax-iinf 4735

This proof depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-int 3971  df-br 4131  df-opab 4193  df-tr 4230  df-id 4438  df-iord 4511  df-on 4513  df-suc 4516  df-iom 4738  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-1o 6687  df-er 6807  df-en 7023  df-fin 7025

This theorem is used by:  vtxlpfi  16531