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

Theorem eq0 4350
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 2157, ax-12 2177. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2110, df-clel 2816. (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 4335 . . 3 ∅ = {𝑦 ∣ ⊥}
21eqeq2i 2750 . 2 (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3 dfcleq 2730 . . 3 (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}))
4 df-clab 2715 . . . . . . 7 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5 sbv 2088 . . . . . . 7 ([𝑥 / 𝑦]⊥ ↔ ⊥)
64, 5bitri 275 . . . . . 6 (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
76bibi2i 337 . . . . 5 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥𝐴 ↔ ⊥))
8 nbfal 1555 . . . . 5 𝑥𝐴 ↔ (𝑥𝐴 ↔ ⊥))
97, 8bitr4i 278 . . . 4 ((𝑥𝐴𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥𝐴)
109albii 1819 . . 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 1538   = wceq 1540  wfal 1552  [wsb 2064  wcel 2108  {cab 2714  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-dif 3954  df-nul 4334
This theorem is referenced by:  neq0  4352  nel0  4354  0el  4363  ssdif0  4366  difin0ss  4373  inssdif0  4374  disjiun  5131  0ex  5307  reldm0  5938  iresn0n0  6072  uzwo  12953  hashgt0elex  14440  nrhmzr  20537  zrninitoringc  20676  hausdiag  23653  rnelfmlem  23960  elons2  28281  prv0  35435  wzel  35825  knoppndv  36535  bj-nul  37057  bj-nuliota  37058  bj-nuliotaALT  37059  nninfnub  37758  prtlem14  38875  orddif0suc  43281
  Copyright terms: Public domain W3C validator