Theorem 1div0apr 30500

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 11948	. . 3 ⊢ / = (𝑥 ∈ ℂ, 𝑦 ∈ (ℂ ∖ {0}) ↦ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥))
2		riotaex 7408	. . 3 ⊢ (℩𝑧 ∈ ℂ (𝑦 · 𝑧) = 𝑥) ∈ V
3	1, 2	dmmpo 8112	. 2 ⊢ dom / = (ℂ × (ℂ ∖ {0}))
4		eqid 2740	. . 3 ⊢ 0 = 0
5		eldifsni 4815	. . . . 5 ⊢ (0 ∈ (ℂ ∖ {0}) → 0 ≠ 0)
6	5	adantl 481	. . . 4 ⊢ ((1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})) → 0 ≠ 0)
7	6	necon2bi 2977	. . 3 ⊢ (0 = 0 → ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0})))
8	4, 7	ax-mp 5	. 2 ⊢ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))
9		ndmovg 7633	. 2 ⊢ ((dom / = (ℂ × (ℂ ∖ {0})) ∧ ¬ (1 ∈ ℂ ∧ 0 ∈ (ℂ ∖ {0}))) → (1 / 0) = ∅)
10	3, 8, 9	mp2an 691	1 ⊢ (1 / 0) = ∅

Syntax hints:  ¬ wn 3   ∧ wa 395   = wceq 1537   ∈ wcel 2108   ≠ wne 2946   ∖ cdif 3973  ∅c0 4352  {csn 4648   × cxp 5698  dom cdm 5700  ℩crio 7403  (class class class)co 7448  ℂcc 11182  0cc0 11184  1c1 11185   · cmul 11189   / cdiv 11947

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-fv 6581  df-riota 7404  df-ov 7451  df-oprab 7452  df-mpo 7453  df-1st 8030  df-2nd 8031  df-div 11948

This theorem is referenced by: (None)