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Theorem 0el 3972
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 3091 . 2 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ∅)
2 eq0 3962 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32rexbii 3070 . 2 (∃𝑥𝐴 𝑥 = ∅ ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
41, 3bitri 264 1 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴𝑦 ¬ 𝑦𝑥)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 196  wal 1521   = wceq 1523  wcel 2030  wrex 2942  c0 3948
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rex 2947  df-v 3233  df-dif 3610  df-nul 3949
This theorem is referenced by:  n0el  3973  axinf2  8575  zfinf2  8577  gneispace  38749
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