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Theorem n0el2 38334
Description: Two ways of expressing that the empty set is not an element of a class. (Contributed by Peter Mazsa, 31-Jan-2018.)
Assertion
Ref Expression
n0el2 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)

Proof of Theorem n0el2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmopab3 5930 . 2 (∀𝑥𝐴𝑦 𝑦𝑥 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = 𝐴)
2 n0el 4364 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑦 𝑦𝑥)
3 cnvepres 38299 . . . 4 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
43dmeqi 5915 . . 3 dom ( E ↾ 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
54eqeq1i 2742 . 2 (dom ( E ↾ 𝐴) = 𝐴 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = 𝐴)
61, 2, 53bitr4i 303 1 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  wral 3061  c0 4333  {copab 5205   E cep 5583  ccnv 5684  dom cdm 5685  cres 5687
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-eprel 5584  df-xp 5691  df-rel 5692  df-cnv 5693  df-dm 5695  df-res 5697
This theorem is referenced by:  n0elim  38651  membpartlem19  38812
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