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Theorem eq0 4344
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 4325 . . 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 4323
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 3952  df-nul 4324
This theorem is referenced by:  neq0  4346  nel0  4351  0el  4361  ssdif0  4364  difin0ss  4369  inssdif0  4370  ralf0OLD  4518  disjiun  5136  0ex  5308  reldm0  5928  iresn0n0  6054  uzwo  12895  hashgt0elex  14361  hausdiag  23149  rnelfmlem  23456  prv0  34421  wzel  34796  knoppndv  35410  bj-nul  35937  bj-nuliota  35938  bj-nuliotaALT  35939  nninfnub  36619  prtlem14  37744  orddif0suc  42018  nrhmzr  46647  zrninitoringc  46969
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