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Theorem disjsuc2 37167
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 37164 . 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 37166 . . . . 5 ((𝑢 ∈ V ∧ 𝐴𝑉) → (([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅ ↔ ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
32el2v1 36991 . . . 4 (𝐴𝑉 → (([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅ ↔ ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
43ralbidv 3178 . . 3 (𝐴𝑉 → (∀𝑢𝐴 ([𝑢](𝑅 E ) ∩ [𝐴](𝑅 E )) = ∅ ↔ ∀𝑢𝐴 ((𝑢𝐴) = ∅ ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
54anbi2d 630 . 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 279 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 397  wo 846   = wceq 1542  wcel 2107  wral 3062  Vcvv 3475  cun 3944  cin 3945  c0 4320  {csn 4624   E cep 5575  ccnv 5671  [cec 8689  cxrn 36948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423  ax-un 7712  ax-reg 9574
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-mpt 5228  df-id 5570  df-eprel 5576  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6487  df-fun 6537  df-fn 6538  df-f 6539  df-fo 6541  df-fv 6543  df-1st 7962  df-2nd 7963  df-ec 8693  df-xrn 37147
This theorem is referenced by:  disjsuc  37535
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