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Theorem eq0f 3917
 Description: The empty set 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 2762 . . 3 𝑥
31, 2cleqf 2787 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
4 noel 3911 . . . 4 ¬ 𝑥 ∈ ∅
54nbn 362 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
65albii 1745 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
73, 6bitr4i 267 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ↔ wb 196  ∀wal 1479   = wceq 1481   ∈ wcel 1988  Ⅎwnfc 2749  ∅c0 3907 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-nul 3908 This theorem is referenced by:  neq0f  3918  eq0  3921  ab0  3942  bnj1476  30891  stoweidlem34  40014
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