Theorem s3iunsndisj 14865

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 865	. . . . 5 ⊢ (𝑎 = 𝑑 → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
2	1	a1d 25	. . . 4 ⊢ (𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)))
3		eliun 4963	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ↔ ∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩})
4		velsn 4607	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} ↔ 𝑠 = ⟨“𝑎𝐵𝑐”⟩)
5		eqeq1 2735	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
6	5	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
7		s3cli 14782	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ⟨“𝑎𝐵𝑐”⟩ ∈ Word V
8		elex 3464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ V)
9		elex 3464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑑 ∈ 𝑌 → 𝑑 ∈ V)
10	9	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌) → 𝑑 ∈ V)
11	8, 10	anim12ci 614	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑑 ∈ V ∧ 𝐵 ∈ V))
12		elex 3464	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ (𝑍 ∖ {𝑑}) → 𝑒 ∈ V)
13	12	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → 𝑒 ∈ V)
14	11, 13	anim12i 613	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → ((𝑑 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑒 ∈ V))
15		df-3an 1089	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V) ↔ ((𝑑 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑒 ∈ V))
16	14, 15	sylibr 233	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V))
17		eqwrds3 14862	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 587	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14792	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎 ∈ V → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎)
20	19	elv 3452	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎
21		simp1 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑)
22	20, 21	eqtr3id 2785	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → 𝑎 = 𝑑)
23	22	adantl 482	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒)) → 𝑎 = 𝑑)
24	18, 23	syl6bi 252	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
25	24	adantr 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
26	6, 25	sylbid 239	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
27	26	ancoms 459	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑠 = ⟨“𝑎𝐵𝑐”⟩ ∧ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})))) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
28	27	con3d 152	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑠 = ⟨“𝑎𝐵𝑐”⟩ ∧ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})))) → (¬ 𝑎 = 𝑑 → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))
29	28	exp32 421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (¬ 𝑎 = 𝑑 → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
30	29	com14 96	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
31	30	imp 407	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
32	31	expd 416	. . . . . . . . . . . . . . . . . . 19 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑐 ∈ (𝑍 ∖ {𝑎}) → (𝑒 ∈ (𝑍 ∖ {𝑑}) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
33	32	com34 91	. . . . . . . . . . . . . . . . . 18 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑐 ∈ (𝑍 ∖ {𝑎}) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
34	33	imp 407	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
35	4, 34	biimtrid 241	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
36	35	imp 407	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))
37	36	imp 407	. . . . . . . . . . . . . 14 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
38		velsn 4607	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩} ↔ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
39	37, 38	sylnibr 328	. . . . . . . . . . . . 13 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
40	39	nrexdv 3142	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
41		eliun 4963	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩} ↔ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
42	40, 41	sylnibr 328	. . . . . . . . . . 11 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
43	42	rexlimdva2 3150	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
44	3, 43	biimtrid 241	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
45	44	ralrimiv 3138	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
46		eqidd 2732	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑑 = 𝑑)
47		eqidd 2732	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝐵 = 𝐵)
48		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑐 = 𝑒)
49	46, 47, 48	s3eqd 14765	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨“𝑑𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩)
50	49	sneqd 4603	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨“𝑑𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑒”⟩})
51	50	cbviunv 5005	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} = ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}
52	51	eleq2i 2824	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
53	52	notbii 319	. . . . . . . . 9 ⊢ (¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
54	53	ralbii 3092	. . . . . . . 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 233	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
56		disj 4412	. . . . . . 7 ⊢ ((∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅ ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
57	55, 56	sylibr 233	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)
58	57	olcd 872	. . . . 5 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
59	58	ex 413	. . . 4 ⊢ (¬ 𝑎 = 𝑑 → ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)))
60	2, 59	pm2.61i 182	. . 3 ⊢ ((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
61	60	ralrimivva 3193	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑌 ∀𝑑 ∈ 𝑌 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
62		sneq 4601	. . . 4 ⊢ (𝑎 = 𝑑 → {𝑎} = {𝑑})
63	62	difeq2d 4087	. . 3 ⊢ (𝑎 = 𝑑 → (𝑍 ∖ {𝑎}) = (𝑍 ∖ {𝑑}))
64		id 22	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑)
65		eqidd 2732	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝐵 = 𝐵)
66		eqidd 2732	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑐 = 𝑐)
67	64, 65, 66	s3eqd 14765	. . . 4 ⊢ (𝑎 = 𝑑 → ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑐”⟩)
68	67	sneqd 4603	. . 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 233	1 ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑌 ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106  ∀wral 3060  ∃wrex 3069  Vcvv 3446   ∖ cdif 3910   ∩ cin 3912  ∅c0 4287  {csn 4591  ∪ ciun 4959  Disj wdisj 5075  ‘cfv 6501  0cc0 11060  1c1 11061  2c2 12217  3c3 12218  ♯chash 14240  Word cword 14414  ⟨“cs3 14743

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2702  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-cnex 11116  ax-resscn 11117  ax-1cn 11118  ax-icn 11119  ax-addcl 11120  ax-addrcl 11121  ax-mulcl 11122  ax-mulrcl 11123  ax-mulcom 11124  ax-addass 11125  ax-mulass 11126  ax-distr 11127  ax-i2m1 11128  ax-1ne0 11129  ax-1rid 11130  ax-rnegex 11131  ax-rrecex 11132  ax-cnre 11133  ax-pre-lttri 11134  ax-pre-lttrn 11135  ax-pre-ltadd 11136  ax-pre-mulgt0 11137

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-nel 3046  df-ral 3061  df-rex 3070  df-rmo 3351  df-reu 3352  df-rab 3406  df-v 3448  df-sbc 3743  df-csb 3859  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-pss 3932  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-tp 4596  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-disj 5076  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-riota 7318  df-ov 7365  df-oprab 7366  df-mpo 7367  df-om 7808  df-1st 7926  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-rdg 8361  df-1o 8417  df-er 8655  df-en 8891  df-dom 8892  df-sdom 8893  df-fin 8894  df-card 9884  df-pnf 11200  df-mnf 11201  df-xr 11202  df-ltxr 11203  df-le 11204  df-sub 11396  df-neg 11397  df-nn 12163  df-2 12225  df-3 12226  df-n0 12423  df-z 12509  df-uz 12773  df-fz 13435  df-fzo 13578  df-hash 14241  df-word 14415  df-concat 14471  df-s1 14496  df-s2 14749  df-s3 14750

This theorem is referenced by:  fusgreghash2wspv  29342