Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjsuc2 Structured version   Visualization version   GIF version

Theorem disjsuc2 37565
Description: Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.)
Assertion
Ref Expression
disjsuc2 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝑅,𝑣   𝑢,𝑉
Allowed substitution hint:   𝑉(𝑣)

Proof of Theorem disjsuc2
StepHypRef Expression
1 disjressuc2 37562 . 2 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅)))
2 disjecxrncnvep 37564 . . . . 5 ((𝑢 ∈ V ∧ 𝐴𝑉) → (([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅ ↔ ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
32el2v1 37389 . . . 4 (𝐴𝑉 → (([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅ ↔ ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
43ralbidv 3176 . . 3 (𝐴𝑉 → (∀𝑢𝐴 ([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅ ↔ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
54anbi2d 628 . 2 (𝐴𝑉 → ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
61, 5bitrd 278 1 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢](𝑅 E ) ∩ [𝑣](𝑅 E )) = ∅) ∧ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395  wo 844   = wceq 1540  wcel 2105  wral 3060  Vcvv 3473  cun 3947  cin 3948  c0 4323  {csn 4629   E cep 5580  ccnv 5676  [cec 8704  cxrn 37346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728  ax-reg 9590
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7978  df-2nd 7979  df-ec 8708  df-xrn 37545
This theorem is referenced by:  disjsuc  37933
  Copyright terms: Public domain W3C validator