Theorem bj-0nelmpt 37618

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 5541	. 2 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		df-mpt 5187	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	2	eqcomi 2774	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	eleq2i 2857	. 2 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵))
5	1, 4	mtbi 325	1 ⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∅c0 4288  {copab 5167   ↦ cmpt 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ne 2961  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-opab 5168  df-mpt 5187

This theorem is referenced by: (None)