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Theorem bj-0nelmpt 35057
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 5462 . 2 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 5152 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32eqcomi 2748 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = (𝑥𝐴𝐵)
43eleq2i 2831 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥𝐴𝐵))
51, 4mtbi 325 1 ¬ ∅ ∈ (𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 399   = wceq 1543  wcel 2112  c0 4253  {copab 5131  cmpt 5151
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2114  ax-9 2122  ax-ext 2710  ax-sep 5208  ax-nul 5215  ax-pr 5338
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2073  df-clab 2717  df-cleq 2731  df-clel 2818  df-ne 2944  df-v 3425  df-dif 3886  df-un 3888  df-nul 4254  df-if 4456  df-sn 4558  df-pr 4560  df-op 4564  df-opab 5132  df-mpt 5152
This theorem is referenced by: (None)
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