Theorem bj-0nelmpt 37559

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 5534	. 2 ⊢ ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
2		df-mpt 5181	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)}
3	2	eqcomi 2770	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} = (𝑥 ∈ 𝐴 ↦ 𝐵)
4	3	eleq2i 2853	. 2 ⊢ (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ 𝐴 ∧ 𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵))
5	1, 4	mtbi 324	1 ⊢ ¬ ∅ ∈ (𝑥 ∈ 𝐴 ↦ 𝐵)

Syntax hints:  ¬ wn 3   ∧ wa 399   = wceq 1559   ∈ wcel 2141  ∅c0 4285  {copab 5161   ↦ cmpt 5180

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733  ax-sep 5245  ax-pr 5389

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-rab 3414  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4480  df-sn 4582  df-pr 4584  df-op 4588  df-opab 5162  df-mpt 5181

This theorem is referenced by: (None)