Theorem 1div0apr 30397

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.

Step	Hyp	Ref	Expression
1		df-div 11836	. . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2		riotaex 7348	. . 3 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
3	1, 2	dmmpo 8050	. 2 ⊢ dom / = (ℂ × (ℂ ∖ {0}))
4		eqid 2729	. . 3 ⊢ 0 = 0
5		eldifsni 4754	. . . . 5 ⊢ (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0)
6	5	adantl 481	. . . 4 ⊢ ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0)
7	6	necon2bi 2955	. . 3 ⊢ (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})))
8	4, 7	ax-mp 5	. 2 ⊢ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
9		ndmovg 7572	. 2 ⊢ ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
10	3, 8, 9	mp2an 692	1 ⊢ (1 / 0) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1540   ∈ wcel 2109   ≠ wne 2925   ∖ cdif 3911  ∅c0 4296  {csn 4589   × cxp 5636  dom cdm 5638  ℩crio 7343  (class class class)co 7387  ℂcc 11066  0cc0 11068  1c1 11069   · cmul 11073   / cdiv 11835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387  ax-un 7711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-sbc 3754  df-csb 3863  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-iun 4957  df-br 5108  df-opab 5170  df-mpt 5189  df-id 5533  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6464  df-fun 6513  df-fn 6514  df-f 6515  df-fv 6519  df-riota 7344  df-ov 7390  df-oprab 7391  df-mpo 7392  df-1st 7968  df-2nd 7969  df-div 11836

This theorem is referenced by: (None)