Theorem bj-0nelmpt 35214

Description: The empty set is not an element of a function (given in maps-to notation). (Contributed by BJ, 30-Dec-2020.)

Assertion

Ref	Expression
bj-0nelmpt	⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)

Proof of Theorem bj-0nelmpt

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		0nelopab 5471	. 2 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		df-mpt 5154	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	2	eqcomi 2747	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	eleq2i 2830	. 2 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵))
5	1, 4	mtbi 321	1 ⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∅c0 4253  {copab 5132   ↦ cmpt 5153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565  df-opab 5133  df-mpt 5154

This theorem is referenced by: (None)