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Theorem eq0 4355
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 2154, ax-12 2174. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2107, df-clel 2813. (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 4340 . . 3 ∅ = {𝑦 ∣ ⊥}
21eqeq2i 2747 . 2 (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3 dfcleq 2727 . . 3 (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}))
4 df-clab 2712 . . . . . . 7 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5 sbv 2085 . . . . . . 7 ([𝑥 / 𝑦]⊥ ↔ ⊥)
64, 5bitri 275 . . . . . 6 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
76bibi2i 337 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥𝐴 ↔ ⊥))
8 nbfal 1551 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴 ↔ ⊥))
97, 8bitr4i 278 . . . 4 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥𝐴)
109albii 1815 . . 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 206  wal 1534   = wceq 1536  wfal 1548  [wsb 2061  wcel 2105  {cab 2711  c0 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-dif 3965  df-nul 4339
This theorem is referenced by:  neq0  4357  nel0  4359  0el  4368  ssdif0  4371  difin0ss  4378  inssdif0  4379  disjiun  5135  0ex  5312  reldm0  5940  iresn0n0  6073  uzwo  12950  hashgt0elex  14436  nrhmzr  20553  zrninitoringc  20692  hausdiag  23668  rnelfmlem  23975  elons2  28295  prv0  35414  wzel  35805  knoppndv  36516  bj-nul  37038  bj-nuliota  37039  bj-nuliotaALT  37040  nninfnub  37737  prtlem14  38855  orddif0suc  43257
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