Theorem eq0 4277

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 2154, ax-12 2171. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2108, df-clel 2816. (Revised by Gino Giotto, 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 4258	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
2	1	eqeq2i 2751	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3		dfcleq 2731	. . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
4		df-clab 2716	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5		sbv 2091	. . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥)
6	4, 5	bitri 274	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
7	6	bibi2i 338	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
8		nbfal 1554	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
9	7, 8	bitr4i 277	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥 ∈ 𝐴)
10	9	albii 1822	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
11	3, 10	bitri 274	. 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
12	2, 11	bitri 274	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 205  ∀wal 1537   = wceq 1539  ⊥wfal 1551  [wsb 2067   ∈ wcel 2106  {cab 2715  ∅c0 4256

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-dif 3890  df-nul 4257

This theorem is referenced by:  neq0  4279  nel0  4284  0el  4294  ssdif0  4297  difin0ss  4302  inssdif0  4303  ralf0OLD  4448  disjiun  5061  0ex  5231  reldm0  5837  iresn0n0  5963  uzwo  12651  hashgt0elex  14116  hausdiag  22796  rnelfmlem  23103  prv0  33392  wzel  33818  knoppndv  34714  bj-nul  35227  bj-nuliota  35228  bj-nuliotaALT  35229  nninfnub  35909  prtlem14  36888  nrhmzr  45431  zrninitoringc  45629