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Theorem eq0f 3879
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 2746 . . 3 𝑥
31, 2cleqf 2771 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
4 noel 3873 . . . 4 ¬ 𝑥 ∈ ∅
54nbn 360 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
65albii 1735 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
73, 6bitr4i 265 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 194  wal 1472   = wceq 1474  wcel 1975  wnfc 2733  c0 3869
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2228  ax-ext 2585
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-clab 2592  df-cleq 2598  df-clel 2601  df-nfc 2735  df-v 3170  df-dif 3538  df-nul 3870
This theorem is referenced by:  neq0f  3880  eq0  3883  ab0  3900  bnj1476  29973  stoweidlem34  38727
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