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Theorem 1div0apr 30306
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) = โˆ…

Proof of Theorem 1div0apr
Dummy variables ๐‘ฅ ๐‘ฆ ๐‘ง are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-div 11912 . . 3 / = (๐‘ฅ โˆˆ โ„‚, ๐‘ฆ โˆˆ (โ„‚ โˆ– {0}) โ†ฆ (โ„ฉ๐‘ง โˆˆ โ„‚ (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ))
2 riotaex 7386 . . 3 (โ„ฉ๐‘ง โˆˆ โ„‚ (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ) โˆˆ V
31, 2dmmpo 8083 . 2 dom / = (โ„‚ ร— (โ„‚ โˆ– {0}))
4 eqid 2728 . . 3 0 = 0
5 eldifsni 4798 . . . . 5 (0 โˆˆ (โ„‚ โˆ– {0}) โ†’ 0 โ‰  0)
65adantl 480 . . . 4 ((1 โˆˆ โ„‚ โˆง 0 โˆˆ (โ„‚ โˆ– {0})) โ†’ 0 โ‰  0)
76necon2bi 2968 . . 3 (0 = 0 โ†’ ยฌ (1 โˆˆ โ„‚ โˆง 0 โˆˆ (โ„‚ โˆ– {0})))
84, 7ax-mp 5 . 2 ยฌ (1 โˆˆ โ„‚ โˆง 0 โˆˆ (โ„‚ โˆ– {0}))
9 ndmovg 7611 . 2 ((dom / = (โ„‚ ร— (โ„‚ โˆ– {0})) โˆง ยฌ (1 โˆˆ โ„‚ โˆง 0 โˆˆ (โ„‚ โˆ– {0}))) โ†’ (1 / 0) = โˆ…)
103, 8, 9mp2an 690 1 (1 / 0) = โˆ…
Colors of variables: wff setvar class
Syntax hints:  ยฌ wn 3   โˆง wa 394   = wceq 1533   โˆˆ wcel 2098   โ‰  wne 2937   โˆ– cdif 3946  โˆ…c0 4326  {csn 4632   ร— cxp 5680  dom cdm 5682  โ„ฉcrio 7381  (class class class)co 7426  โ„‚cc 11146  0cc0 11148  1c1 11149   ยท cmul 11153   / cdiv 11911
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2166  ax-ext 2699  ax-sep 5303  ax-nul 5310  ax-pr 5433  ax-un 7748
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2529  df-eu 2558  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3431  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-iun 5002  df-br 5153  df-opab 5215  df-mpt 5236  df-id 5580  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-iota 6505  df-fun 6555  df-fn 6556  df-f 6557  df-fv 6561  df-riota 7382  df-ov 7429  df-oprab 7430  df-mpo 7431  df-1st 8001  df-2nd 8002  df-div 11912
This theorem is referenced by: (None)
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