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Theorem bj-0nelmpt 36726
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 0nelopab 5569 . 2 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 5233 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32eqcomi 2734 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = (𝑥𝐴𝐵)
43eleq2i 2817 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥𝐴𝐵))
51, 4mtbi 321 1 ¬ ∅ ∈ (𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 394   = wceq 1533  wcel 2098  c0 4322  {copab 5211  cmpt 5232
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463  df-dif 3947  df-un 3949  df-ss 3961  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-opab 5212  df-mpt 5233
This theorem is referenced by: (None)
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