Theorem eq0 4290

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 2811. (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 2809	. 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ ⊥))
3		dfnul4 4275	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
4	3	eqeq2i 2749	. 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 2714  ∅c0 4273

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 2708

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 2715  df-cleq 2728  df-dif 3892  df-nul 4274

This theorem is referenced by:  neq0  4292  nel0  4294  0el  4303  ssdif0  4306  difin0ss  4313  inssdif0  4314  eq0rdv  4347  rzal  4434  ralf0  4437  disjiun  5073  0ex  5242  reldm0  5883  iresn0n0  6019  uzwo  12861  hashgt0elex  14363  nrhmzr  20514  zrninitoringc  20653  hausdiag  23610  rnelfmlem  23917  elons2  28250  prv0  35612  wzel  36004  knoppndv  36794  bj-nul  37363  bj-nuliota  37364  bj-nuliotaALT  37365  nninfnub  38072  prtlem14  39320  orddif0suc  43696