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Theorem s3iunsndisj 13641
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 (𝐵𝑋Disj 𝑎𝑌 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.
StepHypRef Expression
1 orc 400 . . . . 5 (𝑎 = 𝑑 → (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
21a1d 25 . . . 4 (𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) → (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)))
3 eliun 4490 . . . . . . . . . 10 (𝑠 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ↔ ∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩})
4 velsn 4164 . . . . . . . . . . . . . . . . . 18 (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} ↔ 𝑠 = ⟨“𝑎𝐵𝑐”⟩)
5 eqeq1 2625 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
65adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ ↔ ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩))
7 s3cli 13562 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⟨“𝑎𝐵𝑐”⟩ ∈ Word V
8 elex 3198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝐵𝑋𝐵 ∈ V)
9 elex 3198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 (𝑑𝑌𝑑 ∈ V)
109adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ((𝑎𝑌𝑑𝑌) → 𝑑 ∈ V)
118, 10anim12ci 590 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) → (𝑑 ∈ V ∧ 𝐵 ∈ V))
12 elex 3198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝑒 ∈ (𝑍 ∖ {𝑑}) → 𝑒 ∈ V)
1312adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → 𝑒 ∈ V)
1411, 13anim12i 589 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → ((𝑑 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑒 ∈ V))
15 df-3an 1038 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V) ↔ ((𝑑 ∈ V ∧ 𝐵 ∈ V) ∧ 𝑒 ∈ V))
1614, 15sylibr 224 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V))
17 eqwrds3 13638 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((⟨“𝑎𝐵𝑐”⟩ ∈ Word V ∧ (𝑑 ∈ V ∧ 𝐵 ∈ V ∧ 𝑒 ∈ V)) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ ↔ ((#‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒))))
187, 16, 17sylancr 694 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ ↔ ((#‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒))))
19 vex 3189 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝑎 ∈ V
20 s3fv0 13572 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑎 ∈ V → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎)
2119, 20ax-mp 5 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑎
22 simp1 1059 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → (⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑)
2321, 22syl5eqr 2669 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒) → 𝑎 = 𝑑)
2423adantl 482 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((#‘⟨“𝑎𝐵𝑐”⟩) = 3 ∧ ((⟨“𝑎𝐵𝑐”⟩‘0) = 𝑑 ∧ (⟨“𝑎𝐵𝑐”⟩‘1) = 𝐵 ∧ (⟨“𝑎𝐵𝑐”⟩‘2) = 𝑒)) → 𝑎 = 𝑑)
2518, 24syl6bi 243 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
2625adantr 481 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
276, 26sylbid 230 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑}))) ∧ 𝑠 = ⟨“𝑎𝐵𝑐”⟩) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
2827ancoms 469 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑠 = ⟨“𝑎𝐵𝑐”⟩ ∧ ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})))) → (𝑠 = ⟨“𝑑𝐵𝑒”⟩ → 𝑎 = 𝑑))
2928con3d 148 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑠 = ⟨“𝑎𝐵𝑐”⟩ ∧ ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) ∧ (𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})))) → (¬ 𝑎 = 𝑑 → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))
3029exp32 630 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (¬ 𝑎 = 𝑑 → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
3130com14 96 . . . . . . . . . . . . . . . . . . . . . 22 𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
3231imp 445 . . . . . . . . . . . . . . . . . . . . 21 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → ((𝑐 ∈ (𝑍 ∖ {𝑎}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
3332expd 452 . . . . . . . . . . . . . . . . . . . 20 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → (𝑐 ∈ (𝑍 ∖ {𝑎}) → (𝑒 ∈ (𝑍 ∖ {𝑑}) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
3433com34 91 . . . . . . . . . . . . . . . . . . 19 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → (𝑐 ∈ (𝑍 ∖ {𝑎}) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))))
3534imp 445 . . . . . . . . . . . . . . . . . 18 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 = ⟨“𝑎𝐵𝑐”⟩ → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
364, 35syl5bi 232 . . . . . . . . . . . . . . . . 17 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)))
3736imp 445 . . . . . . . . . . . . . . . 16 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → (𝑒 ∈ (𝑍 ∖ {𝑑}) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩))
3837imp 445 . . . . . . . . . . . . . . 15 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
39 velsn 4164 . . . . . . . . . . . . . . 15 (𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩} ↔ 𝑠 = ⟨“𝑑𝐵𝑒”⟩)
4038, 39sylnibr 319 . . . . . . . . . . . . . 14 (((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) ∧ 𝑒 ∈ (𝑍 ∖ {𝑑})) → ¬ 𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
4140nrexdv 2995 . . . . . . . . . . . . 13 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
42 eliun 4490 . . . . . . . . . . . . 13 (𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩} ↔ ∃𝑒 ∈ (𝑍 ∖ {𝑑})𝑠 ∈ {⟨“𝑑𝐵𝑒”⟩})
4341, 42sylnibr 319 . . . . . . . . . . . 12 ((((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) ∧ 𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩}) → ¬ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
4443ex 450 . . . . . . . . . . 11 (((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) ∧ 𝑐 ∈ (𝑍 ∖ {𝑎})) → (𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
4544rexlimdva 3024 . . . . . . . . . 10 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → (∃𝑐 ∈ (𝑍 ∖ {𝑎})𝑠 ∈ {⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
463, 45syl5bi 232 . . . . . . . . 9 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → (𝑠 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} → ¬ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}))
4746ralrimiv 2959 . . . . . . . 8 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → ∀𝑠 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
48 eqidd 2622 . . . . . . . . . . . . . 14 (𝑐 = 𝑒𝑑 = 𝑑)
49 eqidd 2622 . . . . . . . . . . . . . 14 (𝑐 = 𝑒𝐵 = 𝐵)
50 id 22 . . . . . . . . . . . . . 14 (𝑐 = 𝑒𝑐 = 𝑒)
5148, 49, 50s3eqd 13546 . . . . . . . . . . . . 13 (𝑐 = 𝑒 → ⟨“𝑑𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑒”⟩)
5251sneqd 4160 . . . . . . . . . . . 12 (𝑐 = 𝑒 → {⟨“𝑑𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑒”⟩})
5352cbviunv 4525 . . . . . . . . . . 11 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} = 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩}
5453eleq2i 2690 . . . . . . . . . 10 (𝑠 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
5554notbii 310 . . . . . . . . 9 𝑠 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ ¬ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
5655ralbii 2974 . . . . . . . 8 (∀𝑠 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩} ↔ ∀𝑠 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 𝑒 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑒”⟩})
5747, 56sylibr 224 . . . . . . 7 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → ∀𝑠 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
58 disj 3989 . . . . . . 7 (( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅ ↔ ∀𝑠 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ¬ 𝑠 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
5957, 58sylibr 224 . . . . . 6 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)
6059olcd 408 . . . . 5 ((¬ 𝑎 = 𝑑 ∧ (𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌))) → (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
6160ex 450 . . . 4 𝑎 = 𝑑 → ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) → (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅)))
622, 61pm2.61i 176 . . 3 ((𝐵𝑋 ∧ (𝑎𝑌𝑑𝑌)) → (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
6362ralrimivva 2965 . 2 (𝐵𝑋 → ∀𝑎𝑌𝑑𝑌 (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
64 sneq 4158 . . . . 5 (𝑎 = 𝑑 → {𝑎} = {𝑑})
6564difeq2d 3706 . . . 4 (𝑎 = 𝑑 → (𝑍 ∖ {𝑎}) = (𝑍 ∖ {𝑑}))
66 id 22 . . . . . 6 (𝑎 = 𝑑𝑎 = 𝑑)
67 eqidd 2622 . . . . . 6 (𝑎 = 𝑑𝐵 = 𝐵)
68 eqidd 2622 . . . . . 6 (𝑎 = 𝑑𝑐 = 𝑐)
6966, 67, 68s3eqd 13546 . . . . 5 (𝑎 = 𝑑 → ⟨“𝑎𝐵𝑐”⟩ = ⟨“𝑑𝐵𝑐”⟩)
7069sneqd 4160 . . . 4 (𝑎 = 𝑑 → {⟨“𝑎𝐵𝑐”⟩} = {⟨“𝑑𝐵𝑐”⟩})
7165, 70iuneq12d 4512 . . 3 (𝑎 = 𝑑 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} = 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩})
7271disjor 4597 . 2 (Disj 𝑎𝑌 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ↔ ∀𝑎𝑌𝑑𝑌 (𝑎 = 𝑑 ∨ ( 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩} ∩ 𝑐 ∈ (𝑍 ∖ {𝑑}){⟨“𝑑𝐵𝑐”⟩}) = ∅))
7363, 72sylibr 224 1 (𝐵𝑋Disj 𝑎𝑌 𝑐 ∈ (𝑍 ∖ {𝑎}){⟨“𝑎𝐵𝑐”⟩})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3a 1036   = wceq 1480  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  cdif 3552  cin 3554  c0 3891  {csn 4148   ciun 4485  Disj wdisj 4583  cfv 5847  0cc0 9880  1c1 9881  2c2 11014  3c3 11015  #chash 13057  Word cword 13230  ⟨“cs3 13524
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-cnex 9936  ax-resscn 9937  ax-1cn 9938  ax-icn 9939  ax-addcl 9940  ax-addrcl 9941  ax-mulcl 9942  ax-mulrcl 9943  ax-mulcom 9944  ax-addass 9945  ax-mulass 9946  ax-distr 9947  ax-i2m1 9948  ax-1ne0 9949  ax-1rid 9950  ax-rnegex 9951  ax-rrecex 9952  ax-cnre 9953  ax-pre-lttri 9954  ax-pre-lttrn 9955  ax-pre-ltadd 9956  ax-pre-mulgt0 9957
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-nel 2894  df-ral 2912  df-rex 2913  df-reu 2914  df-rmo 2915  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-disj 4584  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-riota 6565  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-fin 7903  df-card 8709  df-pnf 10020  df-mnf 10021  df-xr 10022  df-ltxr 10023  df-le 10024  df-sub 10212  df-neg 10213  df-nn 10965  df-2 11023  df-3 11024  df-n0 11237  df-z 11322  df-uz 11632  df-fz 12269  df-fzo 12407  df-hash 13058  df-word 13238  df-concat 13240  df-s1 13241  df-s2 13530  df-s3 13531
This theorem is referenced by:  fusgreghash2wspv  27057
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