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Theorem eq0 4343
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 2155, ax-12 2172. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2109, df-clel 2811. (Revised by Gino Giotto, 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 4324 . . 3 ∅ = {𝑦 ∣ ⊥}
21eqeq2i 2746 . 2 (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3 dfcleq 2726 . . 3 (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}))
4 df-clab 2711 . . . . . . 7 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5 sbv 2092 . . . . . . 7 ([𝑥 / 𝑦]⊥ ↔ ⊥)
64, 5bitri 275 . . . . . 6 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
76bibi2i 338 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥𝐴 ↔ ⊥))
8 nbfal 1557 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴 ↔ ⊥))
97, 8bitr4i 278 . . . 4 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥𝐴)
109albii 1822 . . 3 (∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥𝐴)
113, 10bitri 275 . 2 (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥 ¬ 𝑥𝐴)
122, 11bitri 275 1 (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wal 1540   = wceq 1542  wfal 1554  [wsb 2068  wcel 2107  {cab 2710  c0 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-dif 3951  df-nul 4323
This theorem is referenced by:  neq0  4345  nel0  4350  0el  4360  ssdif0  4363  difin0ss  4368  inssdif0  4369  ralf0OLD  4517  disjiun  5135  0ex  5307  reldm0  5926  iresn0n0  6052  uzwo  12892  hashgt0elex  14358  hausdiag  23141  rnelfmlem  23448  prv0  34410  wzel  34785  knoppndv  35399  bj-nul  35926  bj-nuliota  35927  bj-nuliotaALT  35928  nninfnub  36608  prtlem14  37733  orddif0suc  42004  nrhmzr  46634  zrninitoringc  46923
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