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Theorem 0el 4320
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 3221 . 2 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ∅)
2 eq0 4303 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32rexbii 3097 . 2 (∃𝑥𝐴 𝑥 = ∅ ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
41, 3bitri 274 1 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1539   = wceq 1541  wcel 2106  wrex 3073  c0 4282
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-rex 3074  df-dif 3913  df-nul 4283
This theorem is referenced by:  n0el  4321  axinf2  9575  zfinf2  9577  gneispace  42387
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