Theorem s3iunsndisj 15001

Description: The union of singletons consisting of length 3 strings which have distinct first and third symbols are disjunct. (Contributed by AV, 17-May-2021.)

Assertion

Ref	Expression
s3iunsndisj	⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑌 ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩})

Distinct variable groups:   𝐵,𝑐   𝑋,𝑐   𝑌,𝑐   𝑍,𝑐   𝐵,𝑎,𝑐   𝑋,𝑎   𝑌,𝑎   𝑍,𝑎

Proof of Theorem s3iunsndisj

Dummy variables 𝑑 𝑒 𝑠 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		orc 880	. . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
2	1	a1d 26	. . . 4 ⊢ (𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)))
3		eliun 4960	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ↔ ∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩})
4		velsn 4605	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} ↔ 𝑠 = ⟨“𝑎𝐵𝑐”⟩)
5		eqeq1 2767	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
6	5	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
7		s3cli 14914	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ⟨“𝑎𝐵𝑐”⟩ ∈ Word V
8		elex 3476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ V)
9		elex 3476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑑 ∈ 𝑌 → 𝑑 ∈ V)
10	9	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌) → 𝑑 ∈ V)
11	8, 10	anim12ci 625	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑑 ∈ V ∧ 𝐵 ∈ V))
12		elex 3476	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ (𝑍 ∖ {𝑑}) → 𝑒 ∈ V)
13	12	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → 𝑒 ∈ V)
14	11, 13	anim12i 624	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → ((𝑑 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑒 ∈ V))
15		df-3an 1105	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V) ↔ ((𝑑 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑒 ∈ V))
16	14, 15	sylibr 237	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V))
17		eqwrds3 14994	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((⟨“𝑎𝐵𝑐”⟩ ∈ Word V ∧ (𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V)) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ ↔ ((♯‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒))))
18	7, 16, 17	sylancr 598	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ ↔ ((♯‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒))))
19		s3fv0 14924	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎 ∈ V → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎)
20	19	elv 3460	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎
21		simp1 1154	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑)
22	20, 21	eqtr3id 2812	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → 𝑎 = 𝑑)
23	22	adantl 486	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒)) → 𝑎 = 𝑑)
24	18, 23	biimtrdi 256	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
25	24	adantr 485	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
26	6, 25	sylbid 243	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
27	26	ancoms 463	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 = ⟨“𝑎𝐵𝑐”⟩ ∧ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})))) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
28	27	con3d 153	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 = ⟨“𝑎𝐵𝑐”⟩ ∧ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})))) → (¬ 𝑎 = 𝑑 → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))
29	28	exp32 425	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (¬ 𝑎 = 𝑑 → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
30	29	com14 97	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
31	30	imp 411	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
32	31	expd 420	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑐 ∈ (𝑍 ∖ {𝑎}) → (𝑒 ∈ (𝑍 ∖ {𝑑}) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
33	32	com34 92	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑐 ∈ (𝑍 ∖ {𝑎}) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
34	33	imp 411	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
35	4, 34	biimtrid 245	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
36	35	imp 411	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))
37	36	imp 411	. . . . . . . . . . . . . 14 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
38		velsn 4605	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩} ↔ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
39	37, 38	sylnibr 332	. . . . . . . . . . . . 13 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
40	39	nrexdv 3160	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
41		eliun 4960	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩} ↔ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
42	40, 41	sylnibr 332	. . . . . . . . . . 11 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
43	42	rexlimdva2 3168	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
44	3, 43	biimtrid 245	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
45	44	ralrimiv 3156	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
46		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑑 = 𝑑)
47		eqidd 2764	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝐵 = 𝐵)
48		id 23	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑐 = 𝑒)
49	46, 47, 48	s3eqd 14897	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨“𝑑𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩)
50	49	sneqd 4601	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨“𝑑𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑒”⟩})
51	50	cbviunv 5003	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} = ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}
52	51	eleq2i 2855	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
53	52	notbii 323	. . . . . . . . 9 ⊢ (¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
54	53	ralbii 3111	. . . . . . . 8 ⊢ (∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
55	45, 54	sylibr 237	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
56		disj 4410	. . . . . . 7 ⊢ ((∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅ ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
57	55, 56	sylibr 237	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)
58	57	olcd 887	. . . . 5 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
59	58	ex 417	. . . 4 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)))
60	2, 59	pm2.61i 184	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
61	60	ralrimivva 3208	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑌 ∀𝑑 ∈ 𝑌 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
62		sneq 4599	. . . 4 ⊢ (𝑎 = 𝑑 → {𝑎} = {𝑑})
63	62	difeq2d 4081	. . 3 ⊢ (𝑎 = 𝑑 → (𝑍 ∖ {𝑎}) = (𝑍 ∖ {𝑑}))
64		id 23	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑)
65		eqidd 2764	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝐵 = 𝐵)
66		eqidd 2764	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑐 = 𝑐)
67	64, 65, 66	s3eqd 14897	. . . 4 ⊢ (𝑎 = 𝑑 → ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑐”⟩)
68	67	sneqd 4601	. . 3 ⊢ (𝑎 = 𝑑 → {⟨“𝑎𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑐”⟩})
69	63, 68	disjiunb 5099	. 2 ⊢ (Disj 𝑎 ∈ 𝑌 ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ↔ ∀𝑎 ∈ 𝑌 ∀𝑑 ∈ 𝑌 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
70	61, 69	sylibr 237	1 ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑌 ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   = wceq 1570   ∈ wcel 2143  ∀wral 3079  ∃wrex 3089  Vcvv 3455   ∖ cdif 3902   ∩ cin 3904  ∅c0 4286  {csn 4589  ∪ ciun 4956  Disj wdisj 5076  ‘cfv 6536  0cc0 11095  1c1 11096  2c2 12290  3c3 12291  ♯chash 14362  Word cword 14546  ⟨“cs3 14875

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pow 5336  ax-pr 5404  ax-un 7732  ax-cnex 11151  ax-resscn 11152  ax-1cn 11153  ax-icn 11154  ax-addcl 11155  ax-addrcl 11156  ax-mulcl 11157  ax-mulrcl 11158  ax-mulcom 11159  ax-addass 11160  ax-mulass 11161  ax-distr 11162  ax-i2m1 11163  ax-1ne0 11164  ax-1rid 11165  ax-rnegex 11166  ax-rrecex 11167  ax-cnre 11168  ax-pre-lttri 11169  ax-pre-lttrn 11170  ax-pre-ltadd 11171  ax-pre-mulgt0 11172

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-nel 3065  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-pss 3925  df-nul 4287  df-if 4488  df-pw 4564  df-sn 4590  df-pr 4592  df-tp 4594  df-op 4596  df-uni 4873  df-int 4913  df-iun 4958  df-disj 5077  df-br 5110  df-opab 5174  df-mpt 5193  df-tr 5219  df-id 5556  df-eprel 5561  df-po 5569  df-so 5570  df-fr 5614  df-we 5616  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-pred 6302  df-ord 6363  df-on 6364  df-lim 6365  df-suc 6366  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-oprab 7414  df-mpo 7415  df-om 7859  df-1st 7982  df-2nd 7983  df-frecs 8274  df-wrecs 8305  df-recs 8354  df-rdg 8393  df-1o 8449  df-er 8690  df-en 8940  df-dom 8941  df-sdom 8942  df-fin 8943  df-card 9921  df-pnf 11240  df-mnf 11241  df-xr 11242  df-ltxr 11243  df-le 11244  df-sub 11438  df-neg 11439  df-nn 12229  df-2 12298  df-3 12299  df-n0 12500  df-z 12587  df-uz 12858  df-fz 13531  df-fzo 13679  df-hash 14363  df-word 14547  df-concat 14604  df-s1 14630  df-s2 14881  df-s3 14882

This theorem is referenced by:  fusgreghash2wspv  30686