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Theorem bj-0nelmpt 32081
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 6473 . 2 ¬ ∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
2 df-mpt 4543 . . . 4 (𝑥𝐴𝐵) = {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)}
32eqcomi 2523 . . 3 {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} = (𝑥𝐴𝐵)
43eleq2i 2584 . 2 (∅ ∈ {⟨𝑥, 𝑦⟩ ∣ (𝑥𝐴𝑦 = 𝐵)} ↔ ∅ ∈ (𝑥𝐴𝐵))
51, 4mtbi 310 1 ¬ ∅ ∈ (𝑥𝐴𝐵)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 382   = wceq 1474  wcel 1938  c0 3777  {copab 4540  cmpt 4541
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-v 3079  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-nul 3778  df-if 3940  df-sn 4029  df-pr 4031  df-op 4035  df-opab 4542  df-mpt 4543
This theorem is referenced by: (None)
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