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Theorem disjlem14 38181
Description: Lemma for disjdmqseq 38188, partim2 38190 and petlem 38195 via disjlem17 38182, (general version of the former prtlem14 38257). (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 38100 . . . 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 3268 . . 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 8743 . . . 4 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
65a1d 25 . . 3 (𝑥 = 𝑦 → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
7 elin 3959 . . . 4 (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) ↔ (𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅))
8 nel02 4327 . . . . 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 242 . . 3 (([𝑥]𝑅 ∩ [𝑦]𝑅) = ∅ → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
116, 10jaoi 854 . 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 844   = wceq 1533  wcel 2098  wral 3055  cin 3942  c0 4317  dom cdm 5669  Rel wrel 5674  [cec 8703   Disj wdisjALTV 37590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8707  df-coss 37794  df-cnvrefrel 37910  df-disjALTV 38088
This theorem is referenced by:  disjlem17  38182
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