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Theorem n0el 33665
 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 2913 . 2 (∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
2 df-ex 1702 . . 3 (∃𝑢 𝑢𝑥 ↔ ¬ ∀𝑢 ¬ 𝑢𝑥)
32ralbii 2976 . 2 (∀𝑥𝐴𝑢 𝑢𝑥 ↔ ∀𝑥𝐴 ¬ ∀𝑢 ¬ 𝑢𝑥)
4 alnex 1703 . . 3 (∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥) ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
5 imnang 1766 . . 3 (∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥) ↔ ∀𝑥 ¬ (𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
6 0el 3921 . . . . 5 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑢 ¬ 𝑢𝑥)
7 df-rex 2914 . . . . 5 (∃𝑥𝐴𝑢 ¬ 𝑢𝑥 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
86, 7bitri 264 . . . 4 (∅ ∈ 𝐴 ↔ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
98notbii 310 . . 3 (¬ ∅ ∈ 𝐴 ↔ ¬ ∃𝑥(𝑥𝐴 ∧ ∀𝑢 ¬ 𝑢𝑥))
104, 5, 93bitr4ri 293 . 2 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥(𝑥𝐴 → ¬ ∀𝑢 ¬ 𝑢𝑥))
111, 3, 103bitr4ri 293 1 (¬ ∅ ∈ 𝐴 ↔ ∀𝑥𝐴𝑢 𝑢𝑥)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384  ∀wal 1478  ∃wex 1701   ∈ wcel 1987  ∀wral 2908  ∃wrex 2909  ∅c0 3897 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-v 3192  df-dif 3563  df-nul 3898 This theorem is referenced by:  prter2  33685
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