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Theorem 1div0apr 28161
Description: Division by zero is forbidden! If we try, we encounter the DO NOT ENTER sign, which in mathematics means it is foolhardy to venture any further, possibly putting the underlying fabric of reality at risk. Based on a dare by David A. Wheeler. (Contributed by Mario Carneiro, 1-Apr-2014.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
1div0apr (1 / 0) = ∅

Proof of Theorem 1div0apr
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-div 11290 . . 3 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2 riotaex 7113 . . 3 (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
31, 2dmmpo 7763 . 2 dom / = (ℂ × (ℂ ∖ {0}))
4 eqid 2825 . . 3 0 = 0
5 eldifsni 4720 . . . . 5 (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0)
65adantl 482 . . . 4 ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0)
76necon2bi 3050 . . 3 (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})))
84, 7ax-mp 5 . 2 ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
9 ndmovg 7324 . 2 ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
103, 8, 9mp2an 688 1 (1 / 0) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 396   = wceq 1530  wcel 2107  wne 3020  cdif 3936  c0 4294  {csn 4563   × cxp 5551  dom cdm 5553  crio 7108  (class class class)co 7151  cc 10527  0cc0 10529  1c1 10530   · cmul 10534   / cdiv 11289
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 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-fv 6359  df-riota 7109  df-ov 7154  df-oprab 7155  df-mpo 7156  df-1st 7683  df-2nd 7684  df-div 11290
This theorem is referenced by: (None)
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