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Theorem disjeccnvep 38188
Description: Property of the epsilon relation. (Contributed by Peter Mazsa, 27-Apr-2020.)
Assertion
Ref Expression
disjeccnvep ((𝐴𝑉𝐵𝑊) → (([𝐴] E ∩ [𝐵] E ) = ∅ ↔ (𝐴𝐵) = ∅))

Proof of Theorem disjeccnvep
StepHypRef Expression
1 eccnvep 38186 . . 3 (𝐴𝑉 → [𝐴] E = 𝐴)
2 eccnvep 38186 . . 3 (𝐵𝑊 → [𝐵] E = 𝐵)
31, 2ineqan12d 4237 . 2 ((𝐴𝑉𝐵𝑊) → ([𝐴] E ∩ [𝐵] E ) = (𝐴𝐵))
43eqeq1d 2736 1 ((𝐴𝑉𝐵𝑊) → (([𝐴] E ∩ [𝐵] E ) = ∅ ↔ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2103  cin 3969  c0 4347   E cep 5602  ccnv 5698  [cec 8757
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2105  ax-9 2113  ax-10 2136  ax-11 2153  ax-12 2173  ax-ext 2705  ax-sep 5320  ax-nul 5327  ax-pr 5450
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3439  df-v 3484  df-dif 3973  df-un 3975  df-in 3977  df-ss 3987  df-nul 4348  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5170  df-opab 5232  df-eprel 5603  df-xp 5705  df-rel 5706  df-cnv 5707  df-dm 5709  df-rn 5710  df-res 5711  df-ima 5712  df-ec 8761
This theorem is referenced by:  disjecxrncnvep  38294
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