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Theorem disjecxrncnvep 38372
Description: Two ways of saying that cosets are disjoint, special case of disjecxrn 38371. (Contributed by Peter Mazsa, 12-Jul-2020.) (Revised by Peter Mazsa, 25-Aug-2023.)
Assertion
Ref Expression
disjecxrncnvep ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅 E ) ∩ [𝐵](𝑅 E )) = ∅ ↔ ((𝐴𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)))

Proof of Theorem disjecxrncnvep
StepHypRef Expression
1 disjecxrn 38371 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅 E ) ∩ [𝐵](𝑅 E )) = ∅ ↔ (([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴] E ∩ [𝐵] E ) = ∅)))
2 orcom 870 . . 3 ((([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅ ∨ ([𝐴] E ∩ [𝐵] E ) = ∅) ↔ (([𝐴] E ∩ [𝐵] E ) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅))
31, 2bitrdi 287 . 2 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅 E ) ∩ [𝐵](𝑅 E )) = ∅ ↔ (([𝐴] E ∩ [𝐵] E ) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)))
4 disjeccnvep 38266 . . 3 ((𝐴𝑉𝐵𝑊) → (([𝐴] E ∩ [𝐵] E ) = ∅ ↔ (𝐴𝐵) = ∅))
54orbi1d 916 . 2 ((𝐴𝑉𝐵𝑊) → ((([𝐴] E ∩ [𝐵] E ) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅) ↔ ((𝐴𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)))
63, 5bitrd 279 1 ((𝐴𝑉𝐵𝑊) → (([𝐴](𝑅 E ) ∩ [𝐵](𝑅 E )) = ∅ ↔ ((𝐴𝐵) = ∅ ∨ ([𝐴]𝑅 ∩ [𝐵]𝑅) = ∅)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  cin 3962  c0 4339   E cep 5588  ccnv 5688  [cec 8742  cxrn 38161
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014  df-ec 8746  df-xrn 38353
This theorem is referenced by:  disjsuc2  38373
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