Theorem eqsndc 7138

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 7014	. . . . . . 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 7001	. . . . 5 ⊢ ((𝐴 ≈ {𝑋} ∧ {𝑋} ≈ 1o) → 𝐴 ≈ 1o)
7	1, 5, 6	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → 𝐴 ≈ 1o)
8		en1 7016	. . . 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 488	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → 𝐴 ⊆ 𝐵)
12		vsnid 3705	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢}
13		eleq2 2295	. . . . . . . 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 3229	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → 𝑢 ∈ 𝐵)
17	2	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → 𝑋 ∈ 𝐵)
18		elssdc.b	. . . . . 6 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
19	18	ad2antrr 488	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦)
20		eqeq1 2238	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
21	20	dcbid 846	. . . . . 6 ⊢ (𝑥 = 𝑢 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑢 = 𝑦))
22		eqeq2 2241	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑢 = 𝑦 ↔ 𝑢 = 𝑋))
23	22	dcbid 846	. . . . . 6 ⊢ (𝑦 = 𝑋 → (DECID 𝑢 = 𝑦 ↔ DECID 𝑢 = 𝑋))
24	21, 23	rspc2va 2925	. . . . 5 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦) → DECID 𝑢 = 𝑋)
25	16, 17, 19, 24	syl21anc 1273	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → DECID 𝑢 = 𝑋)
26		eqeq1 2238	. . . . . . 7 ⊢ (𝐴 = {𝑢} → (𝐴 = {𝑋} ↔ {𝑢} = {𝑋}))
27	26	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (𝐴 = {𝑋} ↔ {𝑢} = {𝑋}))
28		sneqbg 3851	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋))
29	28	elv 2807	. . . . . 6 ⊢ ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋)
30	27, 29	bitrdi 196	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (𝐴 = {𝑋} ↔ 𝑢 = 𝑋))
31	30	dcbid 846	. . . 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 1947	. 2 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → DECID 𝐴 = {𝑋})
34		elssdc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
35		eqeng 6982	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 = {𝑋} → 𝐴 ≈ {𝑋}))
36	34, 35	syl 14	. . . . 5 ⊢ (𝜑 → (𝐴 = {𝑋} → 𝐴 ≈ {𝑋}))
37	36	con3dimp 640	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → ¬ 𝐴 = {𝑋})
38	37	olcd 742	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
39		df-dc 843	. . 3 ⊢ (DECID 𝐴 = {𝑋} ↔ (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
40	38, 39	sylibr 134	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → DECID 𝐴 = {𝑋})
41		snfig 7032	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ Fin)
42	2, 41	syl 14	. . . 4 ⊢ (𝜑 → {𝑋} ∈ Fin)
43		fidcen 7131	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑋} ∈ Fin) → DECID 𝐴 ≈ {𝑋})
44	34, 42, 43	syl2anc 411	. . 3 ⊢ (𝜑 → DECID 𝐴 ≈ {𝑋})
45		exmiddc 844	. . 3 ⊢ (DECID 𝐴 ≈ {𝑋} → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
46	44, 45	syl 14	. 2 ⊢ (𝜑 → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
47	33, 40, 46	mpjaodan 806	1 ⊢ (𝜑 → DECID 𝐴 = {𝑋})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 716  DECID wdc 842   = wceq 1398  ∃wex 1541   ∈ wcel 2202  ∀wral 2511  Vcvv 2803   ⊆ wss 3201  {csn 3673   class class class wbr 4093  1_oc1o 6618   ≈ cen 6950  Fincfn 6952

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 2204  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-nul 4220  ax-pow 4270  ax-pr 4305  ax-un 4536  ax-setind 4641  ax-iinf 4692

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 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-int 3934  df-br 4094  df-opab 4156  df-tr 4193  df-id 4396  df-iord 4469  df-on 4471  df-suc 4474  df-iom 4695  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-1o 6625  df-er 6745  df-en 6953  df-fin 6955

This theorem is referenced by:  vtxlpfi  16214