MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eq0f Structured version   Visualization version   GIF version

Theorem eq0f 4353
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 2903 . . 3 𝑥
31, 2cleqf 2932 . 2 (𝐴 = ∅ ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
4 noel 4344 . . . 4 ¬ 𝑥 ∈ ∅
54nbn 372 . . 3 𝑥𝐴 ↔ (𝑥𝐴𝑥 ∈ ∅))
65albii 1816 . 2 (∀𝑥 ¬ 𝑥𝐴 ↔ ∀𝑥(𝑥𝐴𝑥 ∈ ∅))
73, 6bitr4i 278 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 206  wal 1535   = wceq 1537  wcel 2106  wnfc 2888  c0 4339
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-11 2155  ax-12 2175  ax-ext 2706
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-dif 3966  df-nul 4340
This theorem is referenced by:  neq0f  4354  ab0ALT  4387  bnj1476  34840  stoweidlem34  45990
  Copyright terms: Public domain W3C validator