Theorem disjsuc2 39170

Description: Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.)

Assertion

Ref	Expression
disjsuc2	⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))

Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝑅,𝑣   𝑢,𝑉

Allowed substitution hint:   𝑉(𝑣)

Proof of Theorem disjsuc2

Step	Hyp	Ref	Expression
1		disjressuc2 39167	. 2 ⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝐴](𝑅 ⋉ ◡ E )) = ∅)))
2		disjecxrncnvep 39169	. . . . 5 ⊢ ((𝑢 ∈ V ∧ 𝐴 ∈ 𝑉) → (([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝐴](𝑅 ⋉ ◡ E )) = ∅ ↔ ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
3	2	el2v1 38985	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝐴](𝑅 ⋉ ◡ E )) = ∅ ↔ ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
4	3	ralbidv 3187	. . 3 ⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ 𝐴 ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝐴](𝑅 ⋉ ◡ E )) = ∅ ↔ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
5	4	anbi2d 642	. 2 ⊢ (𝐴 ∈ 𝑉 → ((∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝐴](𝑅 ⋉ ◡ E )) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
6	1, 5	bitrd 282	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ↔ (∀𝑢 ∈ 𝐴 ∀𝑣 ∈ 𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 ⋉ ◡ E ) ∩ [𝑣](𝑅 ⋉ ◡ E )) = ∅) ∧ ∀𝑢 ∈ 𝐴 ((𝑢 ∩ 𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2145  ∀wral 3078  Vcvv 3453   ∪ cun 3900   ∩ cin 3901  ∅c0 4282  {csn 4587   E cep 5558  ^◡ccnv 5658  [cec 8698   ⋉ cxrn 38930

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402  ax-un 7740  ax-reg 9568

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-mpt 5191  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7990  df-2nd 7991  df-ec 8702  df-xrn 39136

This theorem is used by:  disjsuc  39615