Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  disjlem14 Structured version   Visualization version   GIF version

Theorem disjlem14 38270
Description: Lemma for disjdmqseq 38277, partim2 38279 and petlem 38284 via disjlem17 38271, (general version of the former prtlem14 38346). (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 38189 . . . 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 3271 . . 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 8763 . . . 4 (𝑥 = 𝑦 → [𝑥]𝑅 = [𝑦]𝑅)
65a1d 25 . . 3 (𝑥 = 𝑦 → ((𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅) → [𝑥]𝑅 = [𝑦]𝑅))
7 elin 3963 . . . 4 (𝐴 ∈ ([𝑥]𝑅 ∩ [𝑦]𝑅) ↔ (𝐴 ∈ [𝑥]𝑅𝐴 ∈ [𝑦]𝑅))
8 nel02 4333 . . . . 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 856 . 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 846   = wceq 1534  wcel 2099  wral 3058  cin 3946  c0 4323  dom cdm 5678  Rel wrel 5683  [cec 8723   Disj wdisjALTV 37682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ral 3059  df-rex 3068  df-rmo 3373  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-ec 8727  df-coss 37883  df-cnvrefrel 37999  df-disjALTV 38177
This theorem is referenced by:  disjlem17  38271
  Copyright terms: Public domain W3C validator