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Theorem disjressuc2 37247
Description: Double restricted quantification over the union of a set and its singleton. (Contributed by Peter Mazsa, 22-Aug-2023.)
Assertion
Ref Expression
disjressuc2 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
Distinct variable groups:   𝑢,𝐴,𝑣   𝑢,𝑅,𝑣   𝑢,𝑉
Allowed substitution hint:   𝑉(𝑣)

Proof of Theorem disjressuc2
StepHypRef Expression
1 eqeq1 2737 . . . . . 6 (𝑢 = 𝐴 → (𝑢 = 𝑣𝐴 = 𝑣))
2 eceq1 8738 . . . . . . . 8 (𝑢 = 𝐴 → [𝑢]𝑅 = [𝐴]𝑅)
32ineq1d 4211 . . . . . . 7 (𝑢 = 𝐴 → ([𝑢]𝑅 ∩ [𝑣]𝑅) = ([𝐴]𝑅 ∩ [𝑣]𝑅))
43eqeq1d 2735 . . . . . 6 (𝑢 = 𝐴 → (([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅ ↔ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅))
51, 4orbi12d 918 . . . . 5 (𝑢 = 𝐴 → ((𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)))
6 eqeq2 2745 . . . . . 6 (𝑣 = 𝐴 → (𝑢 = 𝑣𝑢 = 𝐴))
7 eceq1 8738 . . . . . . . 8 (𝑣 = 𝐴 → [𝑣]𝑅 = [𝐴]𝑅)
87ineq2d 4212 . . . . . . 7 (𝑣 = 𝐴 → ([𝑢]𝑅 ∩ [𝑣]𝑅) = ([𝑢]𝑅 ∩ [𝐴]𝑅))
98eqeq1d 2735 . . . . . 6 (𝑣 = 𝐴 → (([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅ ↔ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))
106, 9orbi12d 918 . . . . 5 (𝑣 = 𝐴 → ((𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
11 eqeq1 2737 . . . . . 6 (𝑢 = 𝐴 → (𝑢 = 𝐴𝐴 = 𝐴))
122ineq1d 4211 . . . . . . 7 (𝑢 = 𝐴 → ([𝑢]𝑅 ∩ [𝐴]𝑅) = ([𝐴]𝑅 ∩ [𝐴]𝑅))
1312eqeq1d 2735 . . . . . 6 (𝑢 = 𝐴 → (([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅ ↔ ([𝐴]𝑅 ∩ [𝐴]𝑅) = ∅))
1411, 13orbi12d 918 . . . . 5 (𝑢 = 𝐴 → ((𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ↔ (𝐴 = 𝐴 ∨ ([𝐴]𝑅 ∩ [𝐴]𝑅) = ∅)))
155, 10, 142ralunsn 4895 . . . 4 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ∧ (∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ (𝐴 = 𝐴 ∨ ([𝐴]𝑅 ∩ [𝐴]𝑅) = ∅)))))
16 eqid 2733 . . . . . . 7 𝐴 = 𝐴
1716orci 864 . . . . . 6 (𝐴 = 𝐴 ∨ ([𝐴]𝑅 ∩ [𝐴]𝑅) = ∅)
1817biantru 531 . . . . 5 (∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ (𝐴 = 𝐴 ∨ ([𝐴]𝑅 ∩ [𝐴]𝑅) = ∅)))
1918anbi2i 624 . . . 4 (((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)) ↔ ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ∧ (∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ (𝐴 = 𝐴 ∨ ([𝐴]𝑅 ∩ [𝐴]𝑅) = ∅))))
2015, 19bitr4di 289 . . 3 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅))))
21 eqeq1 2737 . . . . . . . . . 10 (𝑢 = 𝑣 → (𝑢 = 𝐴𝑣 = 𝐴))
22 eqcom 2740 . . . . . . . . . 10 (𝑣 = 𝐴𝐴 = 𝑣)
2321, 22bitrdi 287 . . . . . . . . 9 (𝑢 = 𝑣 → (𝑢 = 𝐴𝐴 = 𝑣))
24 eceq1 8738 . . . . . . . . . . . 12 (𝑢 = 𝑣 → [𝑢]𝑅 = [𝑣]𝑅)
2524ineq1d 4211 . . . . . . . . . . 11 (𝑢 = 𝑣 → ([𝑢]𝑅 ∩ [𝐴]𝑅) = ([𝑣]𝑅 ∩ [𝐴]𝑅))
26 incom 4201 . . . . . . . . . . 11 ([𝑣]𝑅 ∩ [𝐴]𝑅) = ([𝐴]𝑅 ∩ [𝑣]𝑅)
2725, 26eqtrdi 2789 . . . . . . . . . 10 (𝑢 = 𝑣 → ([𝑢]𝑅 ∩ [𝐴]𝑅) = ([𝐴]𝑅 ∩ [𝑣]𝑅))
2827eqeq1d 2735 . . . . . . . . 9 (𝑢 = 𝑣 → (([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅ ↔ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅))
2923, 28orbi12d 918 . . . . . . . 8 (𝑢 = 𝑣 → ((𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ↔ (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)))
3029cbvralvw 3235 . . . . . . 7 (∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ↔ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅))
3130biimpi 215 . . . . . 6 (∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) → ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅))
3231pm4.71i 561 . . . . 5 (∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ↔ (∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)))
3332anbi2i 624 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ (∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅))))
34 3anass 1096 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ (∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅))))
35 df-3an 1090 . . . 4 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)) ↔ ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)))
3633, 34, 353bitr2ri 300 . . 3 (((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)) ∧ ∀𝑣𝐴 (𝐴 = 𝑣 ∨ ([𝐴]𝑅 ∩ [𝑣]𝑅) = ∅)) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
3720, 36bitrdi 287 . 2 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))))
38 elneq 9590 . . . . . 6 (𝑢𝐴𝑢𝐴)
3938neneqd 2946 . . . . 5 (𝑢𝐴 → ¬ 𝑢 = 𝐴)
4039biorfd 37086 . . . 4 (𝑢𝐴 → (([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅ ↔ (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
4140ralbiia 3092 . . 3 (∀𝑢𝐴 ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅ ↔ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅))
4241anbi2i 624 . 2 ((∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 (𝑢 = 𝐴 ∨ ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
4337, 42bitr4di 289 1 (𝐴𝑉 → (∀𝑢 ∈ (𝐴 ∪ {𝐴})∀𝑣 ∈ (𝐴 ∪ {𝐴})(𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ↔ (∀𝑢𝐴𝑣𝐴 (𝑢 = 𝑣 ∨ ([𝑢]𝑅 ∩ [𝑣]𝑅) = ∅) ∧ ∀𝑢𝐴 ([𝑢]𝑅 ∩ [𝐴]𝑅) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wral 3062  cun 3946  cin 3947  c0 4322  {csn 4628  [cec 8698
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-12 2172  ax-ext 2704  ax-sep 5299  ax-pr 5427  ax-reg 9584
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-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-br 5149  df-opab 5211  df-xp 5682  df-cnv 5684  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-ec 8702
This theorem is referenced by:  disjsuc2  37250
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