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Theorem disjeccnvep 36733
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 36731 . . 3 (𝐴𝑉 → [𝐴] E = 𝐴)
2 eccnvep 36731 . . 3 (𝐵𝑊 → [𝐵] E = 𝐵)
31, 2ineqan12d 4173 . 2 ((𝐴𝑉𝐵𝑊) → ([𝐴] E ∩ [𝐵] E ) = (𝐴𝐵))
43eqeq1d 2738 1 ((𝐴𝑉𝐵𝑊) → (([𝐴] E ∩ [𝐵] E ) = ∅ ↔ (𝐴𝐵) = ∅))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wcel 2106  cin 3908  c0 4281   E cep 5535  ccnv 5631  [cec 8643
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5255  ax-nul 5262  ax-pr 5383
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ne 2943  df-ral 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-eprel 5536  df-xp 5638  df-rel 5639  df-cnv 5640  df-dm 5642  df-rn 5643  df-res 5644  df-ima 5645  df-ec 8647
This theorem is referenced by:  disjecxrncnvep  36841
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