Theorem eqsndc 7088

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 6966	. . . . . . 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 6953	. . . . 5 ⊢ ((𝐴 ≈ {𝑋} ∧ {𝑋} ≈ 1o) → 𝐴 ≈ 1o)
7	1, 5, 6	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → 𝐴 ≈ 1o)
8		en1 6968	. . . 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 3699	. . . . . . . 8 ⊢ 𝑢 ∈ {𝑢}
13		eleq2 2293	. . . . . . . 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 3226	. . . . 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 2236	. . . . . . 7 ⊢ (𝑥 = 𝑢 → (𝑥 = 𝑦 ↔ 𝑢 = 𝑦))
21	20	dcbid 843	. . . . . 6 ⊢ (𝑥 = 𝑢 → (DECID 𝑥 = 𝑦 ↔ DECID 𝑢 = 𝑦))
22		eqeq2 2239	. . . . . . 7 ⊢ (𝑦 = 𝑋 → (𝑢 = 𝑦 ↔ 𝑢 = 𝑋))
23	22	dcbid 843	. . . . . 6 ⊢ (𝑦 = 𝑋 → (DECID 𝑢 = 𝑦 ↔ DECID 𝑢 = 𝑋))
24	21, 23	rspc2va 2922	. . . . 5 ⊢ (((𝑢 ∈ 𝐵 ∧ 𝑋 ∈ 𝐵) ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 DECID 𝑥 = 𝑦) → DECID 𝑢 = 𝑋)
25	16, 17, 19, 24	syl21anc 1270	. . . 4 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → DECID 𝑢 = 𝑋)
26		eqeq1 2236	. . . . . . 7 ⊢ (𝐴 = {𝑢} → (𝐴 = {𝑋} ↔ {𝑢} = {𝑋}))
27	26	adantl 277	. . . . . 6 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (𝐴 = {𝑋} ↔ {𝑢} = {𝑋}))
28		sneqbg 3844	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋))
29	28	elv 2804	. . . . . 6 ⊢ ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋)
30	27, 29	bitrdi 196	. . . . 5 ⊢ (((𝜑 ∧ 𝐴 ≈ {𝑋}) ∧ 𝐴 = {𝑢}) → (𝐴 = {𝑋} ↔ 𝑢 = 𝑋))
31	30	dcbid 843	. . . 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 1945	. 2 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → DECID 𝐴 = {𝑋})
34		elssdc.a	. . . . . 6 ⊢ (𝜑 → 𝐴 ∈ Fin)
35		eqeng 6934	. . . . . 6 ⊢ (𝐴 ∈ Fin → (𝐴 = {𝑋} → 𝐴 ≈ {𝑋}))
36	34, 35	syl 14	. . . . 5 ⊢ (𝜑 → (𝐴 = {𝑋} → 𝐴 ≈ {𝑋}))
37	36	con3dimp 638	. . . 4 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → ¬ 𝐴 = {𝑋})
38	37	olcd 739	. . 3 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
39		df-dc 840	. . 3 ⊢ (DECID 𝐴 = {𝑋} ↔ (𝐴 = {𝑋} ∨ ¬ 𝐴 = {𝑋}))
40	38, 39	sylibr 134	. 2 ⊢ ((𝜑 ∧ ¬ 𝐴 ≈ {𝑋}) → DECID 𝐴 = {𝑋})
41		snfig 6984	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ Fin)
42	2, 41	syl 14	. . . 4 ⊢ (𝜑 → {𝑋} ∈ Fin)
43		fidcen 7081	. . . 4 ⊢ ((𝐴 ∈ Fin ∧ {𝑋} ∈ Fin) → DECID 𝐴 ≈ {𝑋})
44	34, 42, 43	syl2anc 411	. . 3 ⊢ (𝜑 → DECID 𝐴 ≈ {𝑋})
45		exmiddc 841	. . 3 ⊢ (DECID 𝐴 ≈ {𝑋} → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
46	44, 45	syl 14	. 2 ⊢ (𝜑 → (𝐴 ≈ {𝑋} ∨ ¬ 𝐴 ≈ {𝑋}))
47	33, 40, 46	mpjaodan 803	1 ⊢ (𝜑 → DECID 𝐴 = {𝑋})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 713  DECID wdc 839   = wceq 1395  ∃wex 1538   ∈ wcel 2200  ∀wral 2508  Vcvv 2800   ⊆ wss 3198  {csn 3667   class class class wbr 4086  1_oc1o 6570   ≈ cen 6902  Fincfn 6904

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-nul 4213  ax-pow 4262  ax-pr 4297  ax-un 4528  ax-setind 4633  ax-iinf 4684

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-dif 3200  df-un 3202  df-in 3204  df-ss 3211  df-nul 3493  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-int 3927  df-br 4087  df-opab 4149  df-tr 4186  df-id 4388  df-iord 4461  df-on 4463  df-suc 4466  df-iom 4687  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332  df-1o 6577  df-er 6697  df-en 6905  df-fin 6907

This theorem is referenced by:  vtxlpfi  16096