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Theorem n0el2 35743
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 5756 . 2 (∀𝑥𝐴𝑦 𝑦𝑥 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = 𝐴)
2 n0el 4278 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑦 𝑦𝑥)
3 cnvepres 35708 . . . 4 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
43dmeqi 5741 . . 3 dom ( E ↾ 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
54eqeq1i 2806 . 2 (dom ( E ↾ 𝐴) = 𝐴 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = 𝐴)
61, 2, 53bitr4i 306 1 (¬ ∅ ∈ 𝐴 ↔ dom ( E ↾ 𝐴) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 209  wa 399   = wceq 1538  wex 1781  wcel 2112  wral 3109  c0 4246  {copab 5095   E cep 5432  ccnv 5522  dom cdm 5523  cres 5525
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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-br 5034  df-opab 5096  df-eprel 5433  df-xp 5529  df-rel 5530  df-cnv 5531  df-dm 5533  df-res 5535
This theorem is referenced by:  n0el3  36038
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