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Theorem ab0 4336
Description: The class of sets verifying a property is the empty class if and only if that property is a contradiction. See also abn0 4339 (from which it could be proved using as many essential proof steps but one fewer syntactic step, at the cost of depending on df-ne 3021). (Contributed by BJ, 19-Mar-2021.)
Assertion
Ref Expression
ab0 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)

Proof of Theorem ab0
StepHypRef Expression
1 nfab1 2983 . . 3 𝑥{𝑥𝜑}
21eq0f 4308 . 2 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ {𝑥𝜑})
3 abid 2807 . . . 4 (𝑥 ∈ {𝑥𝜑} ↔ 𝜑)
43notbii 321 . . 3 𝑥 ∈ {𝑥𝜑} ↔ ¬ 𝜑)
54albii 1813 . 2 (∀𝑥 ¬ 𝑥 ∈ {𝑥𝜑} ↔ ∀𝑥 ¬ 𝜑)
62, 5bitri 276 1 ({𝑥𝜑} = ∅ ↔ ∀𝑥 ¬ 𝜑)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 207  wal 1528   = wceq 1530  wcel 2107  {cab 2803  c0 4294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-dif 3942  df-nul 4295
This theorem is referenced by:  dfnf5  4337  rab0  4340  rabeq0  4341  abf  4359
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