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Theorem 0el 4356
Description: Membership of the empty set in another class. (Contributed by NM, 29-Jun-2004.)
Assertion
Ref Expression
0el (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
Distinct variable groups:   𝑥,𝐴   𝑥,𝑦
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem 0el
StepHypRef Expression
1 risset 3225 . 2 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ∅)
2 eq0 4339 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32rexbii 3089 . 2 (∃𝑥𝐴 𝑥 = ∅ ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
41, 3bitri 275 1 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1532   = wceq 1534  wcel 2099  wrex 3065  c0 4318
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2698
This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-rex 3066  df-dif 3947  df-nul 4319
This theorem is referenced by:  n0el  4357  axinf2  9657  zfinf2  9659  gneispace  43536
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