MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  n0el Structured version   Visualization version   GIF version

Theorem n0el 4322
Description: Negated membership of the empty set in another class. (Contributed by Rodolfo Medina, 25-Sep-2010.)
Assertion
Ref Expression
n0el (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑢
Allowed substitution hint:   𝐴(𝑢)

Proof of Theorem n0el
StepHypRef Expression
1 df-ral 3062 . 2 (∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
2 df-ex 1783 . . 3 (∃𝑢 𝑢𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢𝑥)
32ralbii 3093 . 2 (∀𝑥𝐴𝑢 𝑢𝑥 ↔ ∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥)
4 alnex 1784 . . 3 (∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥) ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
5 imnang 1845 . . 3 (∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥) ↔ ∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
6 0el 4321 . . . . 5 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑢 ¬ 𝑢𝑥)
7 df-rex 3071 . . . . 5 (∃𝑥𝐴𝑢 ¬ 𝑢𝑥 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
86, 7bitri 275 . . . 4 (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
98notbii 320 . . 3 (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
104, 5, 93bitr4ri 304 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
111, 3, 103bitr4ri 304 1 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wal 1540  wex 1782  wcel 2107  wral 3061  wrex 3070  c0 4283
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-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-dif 3914  df-nul 4284
This theorem is referenced by:  n0el2  36840  prter2  37389
  Copyright terms: Public domain W3C validator