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Theorem eq0f 4227
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 2899 . . 3 𝑥
31, 2cleqf 2930 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
4 noel 4217 . . . 4 ¬ 𝑥 ∈ ∅
54nbn 376 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
65albii 1826 . 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 1540   = wceq 1542  wcel 2113  wnfc 2879  c0 4209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1916  ax-6 1974  ax-7 2019  ax-8 2115  ax-9 2123  ax-11 2161  ax-12 2178  ax-ext 2710
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2074  df-clab 2717  df-cleq 2730  df-clel 2811  df-nfc 2881  df-dif 3844  df-nul 4210
This theorem is referenced by:  neq0f  4228  eq0OLDOLD  4234  ab0ALT  4262  bnj1476  32390  stoweidlem34  43101
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