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Theorem disjlem14 38799
Description: Lemma for disjdmqseq 38806, partim2 38808 and petlem 38813 via disjlem17 38800, (general version of the former prtlem14 38875). (Contributed by Peter Mazsa, 10-Sep-2021.)
Assertion
Ref Expression
disjlem14 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
Distinct variable group:   𝑥,𝑅,𝑦
Allowed substitution hints:   𝐴(𝑥,𝑦)

Proof of Theorem disjlem14
StepHypRef Expression
1 dfdisjALTV5 38718 . . . 4 ( Disj 𝑅 ↔ (∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) ∧ Rel 𝑅))
21simplbi 497 . . 3 ( Disj 𝑅 → ∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅))
3 rsp2 3277 . . 3 (∀𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅(𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → (𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)))
42, 3syl 17 . 2 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → (𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅)))
5 eceq1 8784 . . . 4 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
65a1d 25 . . 3 (𝑥 = 𝑦 → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
7 elin 3967 . . . 4 (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) ↔ (𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅))
8 nel02 4339 . . . . 5 (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ¬ 𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅))
98pm2.21d 121 . . . 4 (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
107, 9biimtrrid 243 . . 3 (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
116, 10jaoi 858 . 2 ((𝑥 = 𝑦 ∨ ([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
124, 11syl6 35 1 ( Disj 𝑅 → ((𝑥 ∈ dom 𝑅𝑦 ∈ dom 𝑅) → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wo 848   = wceq 1540  wcel 2108  wral 3061  cin 3950  c0 4333  dom cdm 5685  Rel wrel 5690  [cec 8743   Disj wdisjALTV 38216
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ral 3062  df-rex 3071  df-rmo 3380  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-ec 8747  df-coss 38412  df-cnvrefrel 38528  df-disjALTV 38706
This theorem is referenced by:  disjlem17  38800
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