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Theorem eq0f 4278
 Description: A class is equal to the empty set if and only if it has no elements. Theorem 2 of [Suppes] p. 22. (Contributed by BJ, 15-Jul-2021.)
Hypothesis
Ref Expression
eq0f.1 𝑥𝐴
Assertion
Ref Expression
eq0f (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)

Proof of Theorem eq0f
StepHypRef Expression
1 eq0f.1 . . 3 𝑥𝐴
2 nfcv 2974 . . 3 𝑥
31, 2cleqf 3003 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
4 noel 4270 . . . 4 ¬ 𝑥 ∈ ∅
54nbn 376 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
65albii 1821 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
73, 6bitr4i 281 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1536   = wceq 1538   ∈ wcel 2115  Ⅎwnfc 2958  ∅c0 4266 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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-11 2162  ax-12 2178  ax-ext 2793 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-dif 3913  df-nul 4267 This theorem is referenced by:  neq0f  4279  eq0  4281  ab0  4306  bnj1476  32126  stoweidlem34  42469
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