Theorem s3iunsndisj 14941

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 867	. . . . 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 4962	. . . . . . . . . 10 ⊢ (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ↔ ∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩})
4		velsn 4608	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} ↔ 𝑠 = ⟨“𝑎𝐵𝑐”⟩)
5		eqeq1 2734	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
6	5	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
7		s3cli 14854	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ⟨“𝑎𝐵𝑐”⟩ ∈ Word V
8		elex 3471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝐵 ∈ 𝑋 → 𝐵 ∈ V)
9		elex 3471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ (𝑑 ∈ 𝑌 → 𝑑 ∈ V)
10	9	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3471	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (𝑒 ∈ (𝑍 ∖ {𝑑}) → 𝑒 ∈ V)
13	12	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 234	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V))
17		eqwrds3 14934	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 14864	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑎 ∈ V → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎)
20	19	elv 3455	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎
21		simp1 1136	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑)
22	20, 21	eqtr3id 2779	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → 𝑎 = 𝑑)
23	22	adantl 481	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ (((♯‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒)) → 𝑎 = 𝑑)
24	18, 23	biimtrdi 253	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ (((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
25	24	adantr 480	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
26	6, 25	sylbid 240	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
27	26	ancoms 458	. . . . . . . . . . . . . . . . . . . . . . . 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 420	. . . . . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
32	31	expd 415	. . . . . . . . . . . . . . . . . . 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 406	. . . . . . . . . . . . . . . . 17 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
35	4, 34	biimtrid 242	. . . . . . . . . . . . . . . 16 ⊢ (((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
36	35	imp 406	. . . . . . . . . . . . . . 15 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))
37	36	imp 406	. . . . . . . . . . . . . 14 ⊢ (((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
38		velsn 4608	. . . . . . . . . . . . . 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 3129	. . . . . . . . . . . 12 ⊢ ((((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
41		eliun 4962	. . . . . . . . . . . 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 3137	. . . . . . . . . 10 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
44	3, 43	biimtrid 242	. . . . . . . . 9 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
45	44	ralrimiv 3125	. . . . . . . 8 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
46		eqidd 2731	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑑 = 𝑑)
47		eqidd 2731	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝐵 = 𝐵)
48		id 22	. . . . . . . . . . . . . 14 ⊢ (𝑐 = 𝑒 → 𝑐 = 𝑒)
49	46, 47, 48	s3eqd 14837	. . . . . . . . . . . . 13 ⊢ (𝑐 = 𝑒 → ⟨“𝑑𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩)
50	49	sneqd 4604	. . . . . . . . . . . 12 ⊢ (𝑐 = 𝑒 → {⟨“𝑑𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑒”⟩})
51	50	cbviunv 5007	. . . . . . . . . . 11 ⊢ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} = ∪ 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}
52	51	eleq2i 2821	. . . . . . . . . 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 3076	. . . . . . . 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 234	. . . . . . 7 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
56		disj 4416	. . . . . . 7 ⊢ ((∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅ ↔ ∀𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 ∈ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
57	55, 56	sylibr 234	. . . . . 6 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)
58	57	olcd 874	. . . . 5 ⊢ ((¬ 𝑎 = 𝑑 ∧ (𝐵 ∈ 𝑋 ∧ (𝑎 ∈ 𝑌 ∧ 𝑑 ∈ 𝑌))) → (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
59	58	ex 412	. . . 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 3181	. 2 ⊢ (𝐵 ∈ 𝑋 → ∀𝑎 ∈ 𝑌 ∀𝑑 ∈ 𝑌 (𝑎 = 𝑑 ∨ (∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ ∪ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
62		sneq 4602	. . . 4 ⊢ (𝑎 = 𝑑 → {𝑎} = {𝑑})
63	62	difeq2d 4092	. . 3 ⊢ (𝑎 = 𝑑 → (𝑍 ∖ {𝑎}) = (𝑍 ∖ {𝑑}))
64		id 22	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑎 = 𝑑)
65		eqidd 2731	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝐵 = 𝐵)
66		eqidd 2731	. . . . 5 ⊢ (𝑎 = 𝑑 → 𝑐 = 𝑐)
67	64, 65, 66	s3eqd 14837	. . . 4 ⊢ (𝑎 = 𝑑 → ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑐”⟩)
68	67	sneqd 4604	. . 3 ⊢ (𝑎 = 𝑑 → {⟨“𝑎𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑐”⟩})
69	63, 68	disjiunb 5100	. 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 234	1 ⊢ (𝐵 ∈ 𝑋 → Disj 𝑎 ∈ 𝑌 ∪ 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩})

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  ∀wral 3045  ∃wrex 3054  Vcvv 3450   ∖ cdif 3914   ∩ cin 3916  ∅c0 4299  {csn 4592  ∪ ciun 4958  Disj wdisj 5077  ‘cfv 6514  0cc0 11075  1c1 11076  2c2 12248  3c3 12249  ♯chash 14302  Word cword 14485  ⟨“cs3 14815

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2702  ax-rep 5237  ax-sep 5254  ax-nul 5264  ax-pow 5323  ax-pr 5390  ax-un 7714  ax-cnex 11131  ax-resscn 11132  ax-1cn 11133  ax-icn 11134  ax-addcl 11135  ax-addrcl 11136  ax-mulcl 11137  ax-mulrcl 11138  ax-mulcom 11139  ax-addass 11140  ax-mulass 11141  ax-distr 11142  ax-i2m1 11143  ax-1ne0 11144  ax-1rid 11145  ax-rnegex 11146  ax-rrecex 11147  ax-cnre 11148  ax-pre-lttri 11149  ax-pre-lttrn 11150  ax-pre-ltadd 11151  ax-pre-mulgt0 11152

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-nel 3031  df-ral 3046  df-rex 3055  df-rmo 3356  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-pss 3937  df-nul 4300  df-if 4492  df-pw 4568  df-sn 4593  df-pr 4595  df-tp 4597  df-op 4599  df-uni 4875  df-int 4914  df-iun 4960  df-disj 5078  df-br 5111  df-opab 5173  df-mpt 5192  df-tr 5218  df-id 5536  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5594  df-we 5596  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-pred 6277  df-ord 6338  df-on 6339  df-lim 6340  df-suc 6341  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522  df-riota 7347  df-ov 7393  df-oprab 7394  df-mpo 7395  df-om 7846  df-1st 7971  df-2nd 7972  df-frecs 8263  df-wrecs 8294  df-recs 8343  df-rdg 8381  df-1o 8437  df-er 8674  df-en 8922  df-dom 8923  df-sdom 8924  df-fin 8925  df-card 9899  df-pnf 11217  df-mnf 11218  df-xr 11219  df-ltxr 11220  df-le 11221  df-sub 11414  df-neg 11415  df-nn 12194  df-2 12256  df-3 12257  df-n0 12450  df-z 12537  df-uz 12801  df-fz 13476  df-fzo 13623  df-hash 14303  df-word 14486  df-concat 14543  df-s1 14568  df-s2 14821  df-s3 14822

This theorem is referenced by:  fusgreghash2wspv  30271