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Theorem 0el 3920
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 3056 . 2 (∅ ∈ 𝐴 ↔ ∃𝑥𝐴 𝑥 = ∅)
2 eq0 3910 . . 3 (𝑥 = ∅ ↔ ∀𝑦 ¬ 𝑦𝑥)
32rexbii 3035 . 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 1478   = wceq 1480  wcel 1987  wrex 2908  c0 3896
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-rex 2913  df-v 3191  df-dif 3562  df-nul 3897
This theorem is referenced by:  axinf2  8489  zfinf2  8491  n0el  33661  gneispace  37949
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