Theorem eqsndc 7163

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 7037	. . . . . . 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 7024	. . . . 5 |- ( ( A ~~ { X } /\ { X } ~~ 1o ) -> A ~~ 1o )
7	1, 5, 6	syl2anc 411	. . . 4 |- ( ( ph /\ A ~~ { X } ) -> A ~~ 1o )
8		en1 7039	. . . 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 3721	. . . . . . . 8 |- u e. { u }
13		eleq2 2296	. . . . . . . 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 3239	. . . . 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 2239	. . . . . . 7 |- ( x = u -> ( x = y <-> u = y ) )
21	20	dcbid 846	. . . . . 6 |- ( x = u -> (DECID x = y <-> DECID u = y ) )
22		eqeq2 2242	. . . . . . 7 |- ( y = X -> ( u = y <-> u = X ) )
23	22	dcbid 846	. . . . . 6 |- ( y = X -> (DECID u = y <-> DECID u = X ) )
24	21, 23	rspc2va 2935	. . . . 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 1273	. . . 4 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> DECID u = X )
26		eqeq1 2239	. . . . . . 7 |- ( A = { u } -> ( A = { X } <-> { u } = { X } ) )
27	26	adantl 277	. . . . . 6 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> ( A = { X } <-> { u } = { X } ) )
28		sneqbg 3867	. . . . . . 7 |- ( u e. _V -> ( { u } = { X } <-> u = X ) )
29	28	elv 2817	. . . . . 6 |- ( { u } = { X } <-> u = X )
30	27, 29	bitrdi 196	. . . . 5 |- ( ( ( ph /\ A ~~ { X } ) /\ A = { u } ) -> ( A = { X } <-> u = X ) )
31	30	dcbid 846	. . . 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 1948	. 2 |- ( ( ph /\ A ~~ { X } ) -> DECID A = { X } )
34		elssdc.a	. . . . . 6 |- ( ph -> A e. Fin )
35		eqeng 7005	. . . . . 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 742	. . 3 |- ( ( ph /\ -. A ~~ { X } ) -> ( A = { X } \/ -. A = { X } ) )
39		df-dc 843	. . 3 |- (DECID A = { X } <-> ( A = { X } \/ -. A = { X } ) )
40	38, 39	sylibr 134	. 2 |- ( ( ph /\ -. A ~~ { X } ) -> DECID A = { X } )
41		snfig 7056	. . . . 5 |- ( X e. B -> { X } e. Fin )
42	2, 41	syl 14	. . . 4 |- ( ph -> { X } e. Fin )
43		fidcen 7156	. . . 4 |- ( ( A e. Fin /\ { X } e. Fin ) -> DECID A ~~ { X } )
44	34, 42, 43	syl2anc 411	. . 3 |- ( ph -> DECID A ~~ { X } )
45		exmiddc 844	. . 3 |- (DECID A ~~ { X } -> ( A ~~ { X } \/ -. A ~~ { X } ) )
46	44, 45	syl 14	. 2 |- ( ph -> ( A ~~ { X } \/ -. A ~~ { X } ) )
47	33, 40, 46	mpjaodan 806	1 |- ( ph -> DECID A = { X } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ wo 716  DECID wdc 842   = wceq 1398   E.wex 1541   e. wcel 2203   A.wral 2520   _Vcvv 2813   C_ wss 3211   {csn 3689   class class class wbr 4109   1oc1o 6640   ~~ cen 6973   Fincfn 6975

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 2205  ax-14 2206  ax-ext 2214  ax-sep 4228  ax-nul 4236  ax-pow 4287  ax-pr 4322  ax-un 4554  ax-setind 4659  ax-iinf 4710

This theorem depends on definitions:  df-bi 117  df-dc 843  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ne 2413  df-ral 2525  df-rex 2526  df-reu 2527  df-rab 2529  df-v 2815  df-sbc 3043  df-dif 3213  df-un 3215  df-in 3217  df-ss 3224  df-nul 3509  df-pw 3671  df-sn 3695  df-pr 3696  df-op 3698  df-uni 3915  df-int 3950  df-br 4110  df-opab 4172  df-tr 4209  df-id 4414  df-iord 4487  df-on 4489  df-suc 4492  df-iom 4713  df-xp 4755  df-rel 4756  df-cnv 4757  df-co 4758  df-dm 4759  df-rn 4760  df-res 4761  df-ima 4762  df-iota 5312  df-fun 5354  df-fn 5355  df-f 5356  df-f1 5357  df-fo 5358  df-f1o 5359  df-fv 5360  df-1o 6647  df-er 6767  df-en 6976  df-fin 6978

This theorem is referenced by:  vtxlpfi  16285