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Theorem 1div0 11920
Description: You can't divide by zero, because division explicitly excludes zero from the domain of the function. Thus, by the definition of function value, it evaluates to the empty set. (This theorem is for information only and normally is not referenced by other proofs. To be meaningful, it assumes that is not a complex number, which depends on the particular complex number construction that is used.) (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof shortened by SN, 5-Jun-2025.) (New usage is discouraged.)
Assertion
Ref Expression
1div0 (1 / 0) = ∅

Proof of Theorem 1div0
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-div 11919 . . 3 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2 riotaex 7392 . . 3 (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
31, 2dmmpo 8095 . 2 dom / = (ℂ × (ℂ ∖ {0}))
4 neldifsn 4797 . . 3 ¬ 0 ∈ (ℂ ∖ {0})
54intnan 486 . 2 ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
6 ndmovg 7616 . 2 ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
73, 5, 6mp2an 692 1 (1 / 0) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 395   = wceq 1537  wcel 2106  cdif 3960  c0 4339  {csn 4631   × cxp 5687  dom cdm 5689  crio 7387  (class class class)co 7431  cc 11151  0cc0 11153  1c1 11154   · cmul 11158   / cdiv 11918
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fv 6571  df-riota 7388  df-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-div 11919
This theorem is referenced by: (None)
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