Theorem eq0 4304

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 2163, ax-12 2185. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2116, df-clel 2812. (Revised by GG, 6-Sep-2024.)

Assertion

Ref	Expression
eq0	⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Distinct variable group:   𝑥,𝐴

Proof of Theorem eq0

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		biidd 262	. . 3 ⊢ (𝑦 = 𝑥 → (⊥ ↔ ⊥))
2	1	eqabbw 2810	. 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥))
3		dfnul4 4289	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
4	3	eqeq2i 2750	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
5		nbfal 1557	. . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
6	5	albii 1821	. 2 ⊢ (∀𝑥 ¬ 𝑥 ∈ 𝐴 ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥))
7	2, 4, 6	3bitr4i 303	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1540   = wceq 1542  ⊥wfal 1554   ∈ wcel 2114  {cab 2715  ∅c0 4287

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-dif 3906  df-nul 4288

This theorem is referenced by:  neq0  4306  nel0  4308  0el  4317  ssdif0  4320  difin0ss  4327  inssdif0  4328  eq0rdv  4361  rzal  4449  ralf0  4452  disjiun  5088  0ex  5254  reldm0  5885  iresn0n0  6021  uzwo  12836  hashgt0elex  14336  nrhmzr  20482  zrninitoringc  20621  hausdiag  23601  rnelfmlem  23908  elons2  28266  prv0  35643  wzel  36035  knoppndv  36753  bj-nul  37301  bj-nuliota  37302  bj-nuliotaALT  37303  nninfnub  37999  prtlem14  39247  orddif0suc  43622