Theorem 1div0 11798

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.

Step	Hyp	Ref	Expression
1		df-div 11797	. . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2		riotaex 7319	. . 3 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
3	1, 2	dmmpo 8015	. 2 ⊢ dom / = (ℂ × (ℂ ∖ {0}))
4		neldifsn 4736	. . 3 ⊢ ¬ 0 ∈ (ℂ ∖ {0})
5	4	intnan 486	. 2 ⊢ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
6		ndmovg 7541	. 2 ⊢ ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
7	3, 5, 6	mp2an 693	1 ⊢ (1 / 0) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1542   ∈ wcel 2114   ∖ cdif 3887  ∅c0 4274  {csn 4568   × cxp 5620  dom cdm 5622  ℩crio 7314  (class class class)co 7358  ℂcc 11025  0cc0 11027  1c1 11028   · cmul 11032   / cdiv 11796

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5231  ax-nul 5241  ax-pr 5368  ax-un 7680

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-sbc 3730  df-csb 3839  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-iun 4936  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fv 6498  df-riota 7315  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7933  df-2nd 7934  df-div 11797

This theorem is referenced by: (None)