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Theorem 0el 4321
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 3220 . 2 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ∅)
2 eq0 4304 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32rexbii 3094 . 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 1540   = wceq 1542  wcel 2107  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-rex 3071  df-dif 3914  df-nul 4284
This theorem is referenced by:  n0el  4322  axinf2  9581  zfinf2  9583  gneispace  42494
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