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Theorem bj-0nelmpt 33194
 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 ¬ ∅ ∈ (𝑥𝐴𝐵)

Proof of Theorem bj-0nelmpt
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 0neqopab 6740 . 2 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 4763 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32eqcomi 2660 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = (𝑥𝐴𝐵)
43eleq2i 2722 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥𝐴𝐵))
51, 4mtbi 311 1 ¬ ∅ ∈ (𝑥𝐴𝐵)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∅c0 3948  {copab 4745   ↦ cmpt 4762 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-opab 4746  df-mpt 4763 This theorem is referenced by: (None)
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