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Theorem 1div0apr 27294
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 10670 . . 3 / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2 riotaex 6600 . . 3 (𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
31, 2dmmpt2 7225 . 2 dom / = (ℂ × (ℂ ∖ {0}))
4 eqid 2620 . . 3 0 = 0
5 eldifsni 4311 . . . . 5 (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0)
65adantl 482 . . . 4 ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0)
76necon2bi 2821 . . 3 (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})))
84, 7ax-mp 5 . 2 ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
9 ndmovg 6802 . 2 ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
103, 8, 9mp2an 707 1 (1 / 0) = ∅
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wa 384   = wceq 1481  wcel 1988  wne 2791  cdif 3564  c0 3907  {csn 4168   × cxp 5102  dom cdm 5104  crio 6595  (class class class)co 6635  cc 9919  0cc0 9921  1c1 9922   · cmul 9926   / cdiv 10669
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-fv 5884  df-riota 6596  df-ov 6638  df-oprab 6639  df-mpt2 6640  df-1st 7153  df-2nd 7154  df-div 10670
This theorem is referenced by: (None)
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