Theorem eq0 4244

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 2160, ax-12 2177. (Revised by Gino Giotto and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2114, df-clel 2809. (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 4225	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
2	1	eqeq2i 2749	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
3		dfcleq 2729	. . 3 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}))
4		df-clab 2715	. . . . . . 7 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ [𝑥 / 𝑦]⊥)
5		sbv 2096	. . . . . . 7 ⊢ ([𝑥 / 𝑦]⊥ ↔ ⊥)
6	4, 5	bitri 278	. . . . . 6 ⊢ (𝑥 ∈ {𝑦 ∣ ⊥} ↔ ⊥)
7	6	bibi2i 341	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
8		nbfal 1558	. . . . 5 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
9	7, 8	bitr4i 281	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ¬ 𝑥 ∈ 𝐴)
10	9	albii 1827	. . 3 ⊢ (∀𝑥(𝑥 ∈ 𝐴 ↔ 𝑥 ∈ {𝑦 ∣ ⊥}) ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
11	3, 10	bitri 278	. 2 ⊢ (𝐴 = {𝑦 ∣ ⊥} ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)
12	2, 11	bitri 278	1 ⊢ (𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209  ∀wal 1541   = wceq 1543  ⊥wfal 1555  [wsb 2072   ∈ wcel 2112  {cab 2714  ∅c0 4223

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-9 2122  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2715  df-cleq 2728  df-dif 3856  df-nul 4224

This theorem is referenced by:  neq0  4246  nel0  4251  0el  4261  ssdif0  4264  difin0ss  4269  inssdif0  4270  ralf0OLD  4415  disjiun  5026  0ex  5185  reldm0  5782  iresn0n0  5908  uzwo  12472  hashgt0elex  13933  hausdiag  22496  rnelfmlem  22803  prv0  33059  wzel  33498  knoppndv  34400  bj-nul  34913  bj-nuliota  34914  bj-nuliotaALT  34915  nninfnub  35595  prtlem14  36574  nrhmzr  45047  zrninitoringc  45245