Theorem eq0 4302

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 2162, ax-12 2184. (Revised by GG and Steven Nguyen, 28-Jun-2024.) Avoid ax-8 2115, 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 4287	. . 3 ⊢ ∅ = {𝑦 ∣ ⊥}
4	3	eqeq2i 2749	. 2 ⊢ (𝐴 = ∅ ↔ 𝐴 = {𝑦 ∣ ⊥})
5		nbfal 1556	. . 3 ⊢ (¬ 𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ↔ ⊥))
6	5	albii 1820	. 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 1539   = wceq 1541  ⊥wfal 1553   ∈ wcel 2113  {cab 2714  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-9 2123  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-dif 3904  df-nul 4286

This theorem is referenced by:  neq0  4304  nel0  4306  0el  4315  ssdif0  4318  difin0ss  4325  inssdif0  4326  eq0rdv  4359  rzal  4447  ralf0  4450  disjiun  5086  0ex  5252  reldm0  5877  iresn0n0  6013  uzwo  12824  hashgt0elex  14324  nrhmzr  20470  zrninitoringc  20609  hausdiag  23589  rnelfmlem  23896  elons2  28254  prv0  35624  wzel  36016  knoppndv  36734  bj-nul  37257  bj-nuliota  37258  bj-nuliotaALT  37259  nninfnub  37952  prtlem14  39134  orddif0suc  43510