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	⊢ (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 6970	. . . . . . 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 6957	. . . . 5 ⊢ ((𝐴 ≈ {𝑋} ∧ {𝑋} ≈ 1o) → 𝐴 ≈ 1o)
7	1, 5, 6	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → 𝐴 ≈ 1o)
8		en1 6972	. . . 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 3701	. . . . . . . 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 3228	. . . . 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 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 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦) → DECID 𝑢 = 𝑋)
25	16, 17, 19, 24	syl21anc 1272	. . . 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 3846	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋))
29	28	elv 2806	. . . . . 6 ⊢ ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋)
30	27, 29	bitrdi 196	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (𝐴 = {𝑋} ↔ 𝑢 = 𝑋))
31	30	dcbid 845	. . . 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 6938	. . . . . 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 741	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
39		df-dc 842	. . 3 ⊢ (DECID 𝐴 = {𝑋} ↔ (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
40	38, 39	sylibr 134	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → DECID 𝐴 = {𝑋})
41		snfig 6988	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ Fin)
42	2, 41	syl 14	. . . 4 ⊢ (𝜑 → {𝑋} ∈ Fin)
43		fidcen 7087	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑋} ∈ Fin) → DECID 𝐴 ≈ {𝑋})
44	34, 42, 43	syl2anc 411	. . 3 ⊢ (𝜑 → DECID 𝐴 ≈ {𝑋})
45		exmiddc 843	. . 3 ⊢ (DECID 𝐴 ≈ {𝑋} → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
46	44, 45	syl 14	. 2 ⊢ (𝜑 → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
47	33, 40, 46	mpjaodan 805	1 ⊢ (𝜑 → DECID 𝐴 = {𝑋})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 715  DECID wdc 841   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  Vcvv 2802   ⊆ wss 3200  {csn 3669   class class class wbr 4088  1_oc1o 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