Theorem eq0 4373

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 2110, df-clel 2819. (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 4354	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
2	1	eqeq2i 2753	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3		dfcleq 2733	. . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
4		df-clab 2718	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5		sbv 2088	. . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥)
6	4, 5	bitri 275	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
7	6	bibi2i 337	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
8		nbfal 1552	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
9	7, 8	bitr4i 278	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥 ∈ 𝐴)
10	9	albii 1817	. . 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 1535   = wceq 1537  ⊥wfal 1549  [wsb 2064   ∈ wcel 2108  {cab 2717  ∅c0 4352

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-dif 3979  df-nul 4353

This theorem is referenced by:  neq0  4375  nel0  4377  0el  4386  ssdif0  4389  difin0ss  4396  inssdif0  4397  disjiun  5154  0ex  5325  reldm0  5952  iresn0n0  6083  uzwo  12976  hashgt0elex  14450  nrhmzr  20563  zrninitoringc  20698  hausdiag  23674  rnelfmlem  23981  elons2  28299  prv0  35398  wzel  35788  knoppndv  36500  bj-nul  37022  bj-nuliota  37023  bj-nuliotaALT  37024  nninfnub  37711  prtlem14  38830  orddif0suc  43230