Theorem s3iunsndisj 14677

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 864	. . . . 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 4934	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ↔ ∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩})
4		velsn 4583	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} ↔ 𝑠 = ⟨“𝑎𝐵𝑐”⟩)
5		eqeq1 2744	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 14592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ⟨“𝑎𝐵𝑐”⟩ ∈ Word V
8		elex 3449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ V)
9		elex 3449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3449	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1088	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14674	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14602	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎 ∈ V → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎)
20	19	elv 3437	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎
21		simp1 1135	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑)
22	20, 21	eqtr3id 2794	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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	syl5bi 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 4583	. . . . . . . . . . . . . 14 ⊢ (𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩} ↔ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
39	37, 38	sylnibr 329	. . . . . . . . . . . . 13 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
40	39	nrexdv 3200	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
41		eliun 4934	. . . . . . . . . . . 12 ⊢ (𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩} ↔ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
42	40, 41	sylnibr 329	. . . . . . . . . . 11 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
43	42	rexlimdva2 3218	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
44	3, 43	syl5bi 241	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
45	44	ralrimiv 3109	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
46		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑑 = 𝑑)
47		eqidd 2741	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝐵 = 𝐵)
48		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑐 = 𝑒)
49	46, 47, 48	s3eqd 14575	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨“𝑑𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩)
50	49	sneqd 4579	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨“𝑑𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑒”⟩})
51	50	cbviunv 4975	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} = ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}
52	51	eleq2i 2832	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
53	52	notbii 320	. . . . . . . . 9 ⊢ (¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
54	53	ralbii 3093	. . . . . . . 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 4387	. . . . . . 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 871	. . . . 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 3117	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑌 ∀𝑑 ∈ 𝑌 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
62		sneq 4577	. . . 4 ⊢ (𝑎 = 𝑑 → {𝑎} = {𝑑})
63	62	difeq2d 4062	. . 3 ⊢ (𝑎 = 𝑑 → (𝑍 ∖ {𝑎}) = (𝑍 ∖ {𝑑}))
64		id 22	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑)
65		eqidd 2741	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝐵 = 𝐵)
66		eqidd 2741	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑐 = 𝑐)
67	64, 65, 66	s3eqd 14575	. . . 4 ⊢ (𝑎 = 𝑑 → ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑐”⟩)
68	67	sneqd 4579	. . 3 ⊢ (𝑎 = 𝑑 → {⟨“𝑎𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑐”⟩})
69	63, 68	disjiunb 5068	. 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 844   ∧ w3a 1086   = wceq 1542   ∈ wcel 2110  ∀wral 3066  ∃wrex 3067  Vcvv 3431   ∖ cdif 3889   ∩ cin 3891  ∅c0 4262  {csn 4567  ∪ ciun 4930  Disj wdisj 5044  ‘cfv 6432  0cc0 10872  1c1 10873  2c2 12028  3c3 12029  ♯chash 14042  Word cword 14215  ⟨“cs3 14553

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-rep 5214  ax-sep 5227  ax-nul 5234  ax-pow 5292  ax-pr 5356  ax-un 7582  ax-cnex 10928  ax-resscn 10929  ax-1cn 10930  ax-icn 10931  ax-addcl 10932  ax-addrcl 10933  ax-mulcl 10934  ax-mulrcl 10935  ax-mulcom 10936  ax-addass 10937  ax-mulass 10938  ax-distr 10939  ax-i2m1 10940  ax-1ne0 10941  ax-1rid 10942  ax-rnegex 10943  ax-rrecex 10944  ax-cnre 10945  ax-pre-lttri 10946  ax-pre-lttrn 10947  ax-pre-ltadd 10948  ax-pre-mulgt0 10949

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-nel 3052  df-ral 3071  df-rex 3072  df-reu 3073  df-rmo 3074  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-tp 4572  df-op 4574  df-uni 4846  df-int 4886  df-iun 4932  df-disj 5045  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-riota 7228  df-ov 7274  df-oprab 7275  df-mpo 7276  df-om 7707  df-1st 7824  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193  df-rdg 8232  df-1o 8288  df-er 8481  df-en 8717  df-dom 8718  df-sdom 8719  df-fin 8720  df-card 9698  df-pnf 11012  df-mnf 11013  df-xr 11014  df-ltxr 11015  df-le 11016  df-sub 11207  df-neg 11208  df-nn 11974  df-2 12036  df-3 12037  df-n0 12234  df-z 12320  df-uz 12582  df-fz 13239  df-fzo 13382  df-hash 14043  df-word 14216  df-concat 14272  df-s1 14299  df-s2 14559  df-s3 14560

This theorem is referenced by:  fusgreghash2wspv  28695