MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eq0 Structured version   Visualization version   GIF version

Theorem eq0 4345
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 NM, 29-Aug-1993.) Avoid ax-11 2147, ax-12 2167. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2101, df-clel 2803. (Revised by GG, 6-Sep-2024.)
Assertion
Ref Expression
eq0 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Distinct variable group:   𝑥,𝐴

Proof of Theorem eq0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 dfnul4 4326 . . 3 ∅ = {𝑦 ∣ ⊥}
21eqeq2i 2739 . 2 (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3 dfcleq 2719 . . 3 (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}))
4 df-clab 2704 . . . . . . 7 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5 sbv 2084 . . . . . . 7 ([𝑥 / 𝑦]⊥ ↔ ⊥)
64, 5bitri 274 . . . . . 6 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
76bibi2i 336 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥𝐴 ↔ ⊥))
8 nbfal 1549 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴 ↔ ⊥))
97, 8bitr4i 277 . . . 4 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥𝐴)
109albii 1814 . . 3 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥𝐴)
113, 10bitri 274 . 2 (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥 ¬ 𝑥𝐴)
122, 11bitri 274 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1532   = wceq 1534  wfal 1546  [wsb 2060  wcel 2099  {cab 2703  c0 4324
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-9 2109  ax-ext 2697
This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-dif 3951  df-nul 4325
This theorem is referenced by:  neq0  4347  nel0  4349  0el  4358  ssdif0  4361  difin0ss  4368  inssdif0  4369  ralf0OLD  4514  disjiun  5133  0ex  5304  reldm0  5926  iresn0n0  6055  uzwo  12940  hashgt0elex  14412  nrhmzr  20514  zrninitoringc  20649  hausdiag  23636  rnelfmlem  23943  elons2  28248  prv0  35270  wzel  35660  knoppndv  36249  bj-nul  36775  bj-nuliota  36776  bj-nuliotaALT  36777  nninfnub  37464  prtlem14  38584  orddif0suc  42970
  Copyright terms: Public domain W3C validator