Theorem eqsndc 7076

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 6957	. . . . . . 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 6944	. . . . 5 ⊢ ((𝐴 ≈ {𝑋} ∧ {𝑋} ≈ 1o) → 𝐴 ≈ 1o)
7	1, 5, 6	syl2anc 411	. . . 4 ⊢ ((𝜑 ∧ 𝐴 ≈ {𝑋}) → 𝐴 ≈ 1o)
8		en1 6959	. . . 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 3698	. . . . . . . 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 3225	. . . . 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 2921	. . . . 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 3841	. . . . . . 7 ⊢ (𝑢 ∈ V → ({𝑢} = {𝑋} ↔ 𝑢 = 𝑋))
29	28	elv 2803	. . . . . 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 6925	. . . . . 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 6975	. . . . 5 ⊢ (𝑋 ∈ 𝐵 → {𝑋} ∈ Fin)
42	2, 41	syl 14	. . . 4 ⊢ (𝜑 → {𝑋} ∈ Fin)
43		fidcen 7069	. . . 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 2799   ⊆ wss 3197  {csn 3666   class class class wbr 4083  1_oc1o 6561   ≈ cen 6893  Fincfn 6895

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 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

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 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-tr 4183  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1o 6568  df-er 6688  df-en 6896  df-fin 6898

This theorem is referenced by:  vtxlpfi  16049