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	⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
elssdc.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
elssdc.ss	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
elssdc.a	⊢ (𝜑 → 𝐴 ∈ Fin)

Assertion

Ref	Expression
eqsndc	⊢ (𝜑 → DECID 𝐴 = {𝑋})

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

Allowed substitution hints:   𝜑(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 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → 𝐴 ≈ {𝑋})
2		elssdc.x	. . . . . . 7 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
3		ensn1g 7084	. . . . . . 7 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ≈ 1o)
4	2, 3	syl 14	. . . . . 6 ⊢ (𝜑 → {𝑋} ≈ 1o)
5	4	adantr 276	. . . . 5 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → {𝑋} ≈ 1o)
6		entr 7071	. . . . 5 ⊢ ((𝐴 ≈ {𝑋} ∧ {𝑋} ≈ 1o) → 𝐴 ≈ 1o)
7	1, 5, 6	syl2anc 415	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → 𝐴 ≈ 1o)
8		en1 7086	. . . 4 ⊢ (𝐴 ≈ 1o ↔ ∃𝑢 𝐴 = {𝑢})
9	7, 8	sylib 122	. . 3 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → ∃𝑢 𝐴 = {𝑢})
10		elssdc.ss	. . . . . . 7 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
11	10	ad2antrr 492	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → 𝐴 ⊆ 𝐵)
12		vsnid 3741	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢}
13		eleq2 2302	. . . . . . . 8 ⊢ (𝐴 = {𝑢} → (𝑢 ∈ 𝐴 ↔ 𝑢 ∈ {𝑢}))
14	12, 13	mpbiri 168	. . . . . . 7 ⊢ (𝐴 = {𝑢} → 𝑢 ∈ 𝐴)
15	14	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → 𝑢 ∈ 𝐴)
16	11, 15	sseldd 3249	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → 𝑢 ∈ 𝐵)
17	2	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → 𝑋 ∈ 𝐵)
18		elssdc.b	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
19	18	ad2antrr 492	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 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 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦) → DECID 𝑢 = 𝑋)
25	16, 17, 19, 24	syl21anc 1277	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → DECID 𝑢 = 𝑋)
26		eqeq1 2245	. . . . . . 7 ⊢ (𝐴 = {𝑢} → (𝐴 = {𝑋} ↔ {𝑢} = {𝑋}))
27	26	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (𝐴 = {𝑋} ↔ {𝑢} = {𝑋}))
28		sneqbg 3888	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋))
29	28	elv 2825	. . . . . 6 ⊢ ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋)
30	27, 29	bitrdi 196	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (𝐴 = {𝑋} ↔ 𝑢 = 𝑋))
31	30	dcbid 850	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (DECID 𝐴 = {𝑋} ↔ DECID 𝑢 = 𝑋))
32	25, 31	mpbird 167	. . 3 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → DECID 𝐴 = {𝑋})
33	9, 32	exlimddv 1954	. 2 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → DECID 𝐴 = {𝑋})
34		elssdc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
35		eqeng 7052	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 = {𝑋} → 𝐴 ≈ {𝑋}))
36	34, 35	syl 14	. . . . 5 ⊢ (𝜑 → (𝐴 = {𝑋} → 𝐴 ≈ {𝑋}))
37	36	con3dimp 644	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → ¬ 𝐴 = {𝑋})
38	37	olcd 746	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
39		df-dc 847	. . 3 ⊢ (DECID 𝐴 = {𝑋} ↔ (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
40	38, 39	sylibr 134	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → DECID 𝐴 = {𝑋})
41		snfig 7103	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ Fin)
42	2, 41	syl 14	. . . 4 ⊢ (𝜑 → {𝑋} ∈ Fin)
43		fidcen 7203	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑋} ∈ Fin) → DECID 𝐴 ≈ {𝑋})
44	34, 42, 43	syl2anc 415	. . 3 ⊢ (𝜑 → DECID 𝐴 ≈ {𝑋})
45		exmiddc 848	. . 3 ⊢ (DECID 𝐴 ≈ {𝑋} → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
46	44, 45	syl 14	. 2 ⊢ (𝜑 → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
47	33, 40, 46	mpjaodan 810	1 ⊢ (𝜑 → DECID 𝐴 = {𝑋})

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720  DECID wdc 846   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  Vcvv 2821   ⊆ wss 3220  {csn 3709   class class class wbr 4130  1_oc1o 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