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Theorem n0el 4318
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 3140 . 2 (∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
2 df-ex 1772 . . 3 (∃𝑢 𝑢𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢𝑥)
32ralbii 3162 . 2 (∀𝑥𝐴𝑢 𝑢𝑥 ↔ ∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥)
4 alnex 1773 . . 3 (∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥) ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
5 imnang 1833 . . 3 (∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥) ↔ ∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
6 0el 4317 . . . . 5 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑢 ¬ 𝑢𝑥)
7 df-rex 3141 . . . . 5 (∃𝑥𝐴𝑢 ¬ 𝑢𝑥 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
86, 7bitri 276 . . . 4 (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
98notbii 321 . . 3 (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
104, 5, 93bitr4ri 305 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
111, 3, 103bitr4ri 305 1 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wal 1526  wex 1771  wcel 2105  wral 3135  wrex 3136  c0 4288
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-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-dif 3936  df-nul 4289
This theorem is referenced by:  n0el2  35471  prter2  35897
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