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Theorem n0el 4293
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 3135 . 2 (∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
2 df-ex 1782 . . 3 (∃𝑢 𝑢𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢𝑥)
32ralbii 3157 . 2 (∀𝑥𝐴𝑢 𝑢𝑥 ↔ ∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥)
4 alnex 1783 . . 3 (∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥) ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
5 imnang 1843 . . 3 (∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥) ↔ ∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
6 0el 4292 . . . . 5 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑢 ¬ 𝑢𝑥)
7 df-rex 3136 . . . . 5 (∃𝑥𝐴𝑢 ¬ 𝑢𝑥 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
86, 7bitri 278 . . . 4 (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
98notbii 323 . . 3 (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
104, 5, 93bitr4ri 307 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
111, 3, 103bitr4ri 307 1 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wal 1536  wex 1781  wcel 2114  wral 3130  wrex 3131  c0 4265
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 2116  ax-9 2124  ax-11 2161  ax-12 2178  ax-ext 2794
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-ral 3135  df-rex 3136  df-dif 3911  df-nul 4266
This theorem is referenced by:  n0el2  35712  prter2  36139
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