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Theorem n0el2 35471
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 5781 . 2 (∀𝑥𝐴𝑦 𝑦𝑥 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = 𝐴)
2 n0el 4318 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑦 𝑦𝑥)
3 cnvepres 35436 . . . 4 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
43dmeqi 5766 . . 3 dom ( E ↾ 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
54eqeq1i 2823 . 2 (dom ( E ↾ 𝐴) = 𝐴 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = 𝐴)
61, 2, 53bitr4i 304 1 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wa 396   = wceq 1528  wex 1771  wcel 2105  wral 3135  c0 4288  {copab 5119   E cep 5457  ccnv 5547  dom cdm 5548  cres 5550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-br 5058  df-opab 5120  df-eprel 5458  df-xp 5554  df-rel 5555  df-cnv 5556  df-dm 5558  df-res 5560
This theorem is referenced by:  n0el3  35765
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