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Theorem 1div0apr 30230
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 11876 . . 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 7365 . . 3 (โ„ฉ๐‘ง โˆˆ โ„‚ (๐‘ฆ ยท ๐‘ง) = ๐‘ฅ) โˆˆ V
31, 2dmmpo 8056 . 2 dom / = (โ„‚ ร— (โ„‚ โˆ– {0}))
4 eqid 2726 . . 3 0 = 0
5 eldifsni 4788 . . . . 5 (0 โˆˆ (โ„‚ โˆ– {0}) โ†’ 0 โ‰  0)
65adantl 481 . . . 4 ((1 โˆˆ โ„‚ โˆง 0 โˆˆ (โ„‚ โˆ– {0})) โ†’ 0 โ‰  0)
76necon2bi 2965 . . 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 7587 . 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 689 1 (1 / 0) = โˆ…
Colors of variables: wff setvar class
Syntax hints:  ยฌ wn 3   โˆง wa 395   = wceq 1533   โˆˆ wcel 2098   โ‰  wne 2934   โˆ– cdif 3940  โˆ…c0 4317  {csn 4623   ร— cxp 5667  dom cdm 5669  โ„ฉcrio 7360  (class class class)co 7405  โ„‚cc 11110  0cc0 11112  1c1 11113   ยท cmul 11117   / cdiv 11875
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 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-riota 7361  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7974  df-2nd 7975  df-div 11876
This theorem is referenced by: (None)
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