Theorem eq0 4313

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 2158, ax-12 2178. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2111, df-clel 2803. (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		dfnul4 4298	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
2	1	eqeq2i 2742	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3		dfcleq 2722	. . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
4		df-clab 2708	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5		sbv 2089	. . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥)
6	4, 5	bitri 275	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
7	6	bibi2i 337	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
8		nbfal 1555	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
9	7, 8	bitr4i 278	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥 ∈ 𝐴)
10	9	albii 1819	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
11	3, 10	bitri 275	. 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
12	2, 11	bitri 275	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206  ∀wal 1538   = wceq 1540  ⊥wfal 1552  [wsb 2065   ∈ wcel 2109  {cab 2707  ∅c0 4296

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 2008  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-dif 3917  df-nul 4297

This theorem is referenced by:  neq0  4315  nel0  4317  0el  4326  ssdif0  4329  difin0ss  4336  inssdif0  4337  disjiun  5095  0ex  5262  reldm0  5891  iresn0n0  6025  uzwo  12870  hashgt0elex  14366  nrhmzr  20446  zrninitoringc  20585  hausdiag  23532  rnelfmlem  23839  elons2  28159  prv0  35417  wzel  35812  knoppndv  36522  bj-nul  37044  bj-nuliota  37045  bj-nuliotaALT  37046  nninfnub  37745  prtlem14  38867  orddif0suc  43257