Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  n0el2 Structured version   Visualization version   GIF version

Theorem n0el2 37202
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 5920 . 2 (∀𝑥𝐴𝑦 𝑦𝑥 ↔ dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)} = 𝐴)
2 n0el 4362 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑦 𝑦𝑥)
3 cnvepres 37167 . . . 4 ( E ↾ 𝐴) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
43dmeqi 5905 . . 3 dom ( E ↾ 𝐴) = dom {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦𝑥)}
54eqeq1i 2738 . 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 205  wa 397   = wceq 1542  wex 1782  wcel 2107  wral 3062  c0 4323  {copab 5211   E cep 5580  ccnv 5676  dom cdm 5677  cres 5679
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-res 5689
This theorem is referenced by:  n0elim  37520  membpartlem19  37681
  Copyright terms: Public domain W3C validator