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Theorem 1div0 10724
 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.) (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 10723 . . 3 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2 riotaex 6655 . . 3 (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
31, 2dmmpt2 7285 . 2 dom / = (ℂ × (ℂ ∖ {0}))
4 eqid 2651 . . 3 0 = 0
5 eldifsni 4353 . . . . 5 (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0)
65adantl 481 . . . 4 ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0)
76necon2bi 2853 . . 3 (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})))
84, 7ax-mp 5 . 2 ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
9 ndmovg 6859 . 2 ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
103, 8, 9mp2an 708 1 (1 / 0) = ∅
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   ∧ wa 383   = wceq 1523   ∈ wcel 2030   ≠ wne 2823   ∖ cdif 3604  ∅c0 3948  {csn 4210   × cxp 5141  dom cdm 5143  ℩crio 6650  (class class class)co 6690  ℂcc 9972  0cc0 9974  1c1 9975   · cmul 9979   / cdiv 10722 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-8 2032  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-pow 4873  ax-pr 4936  ax-un 6991 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-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  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-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fv 5934  df-riota 6651  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-1st 7210  df-2nd 7211  df-div 10723 This theorem is referenced by: (None)
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