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Theorem 1div0apr 21762
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.
StepHypRef Expression
1 df-div 9678 . . 3  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC ( y  x.  z )  =  x ) )
2 riotaex 6553 . . 3  |-  ( iota_ z  e.  CC ( y  x.  z )  =  x )  e.  _V
31, 2dmmpt2 6421 . 2  |-  dom  /  =  ( CC  X.  ( CC  \  { 0 } ) )
4 eqid 2436 . . 3  |-  0  =  0
5 eldifsni 3928 . . . . 5  |-  ( 0  e.  ( CC  \  { 0 } )  ->  0  =/=  0
)
65adantl 453 . . . 4  |-  ( ( 1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )  ->  0  =/=  0 )
76necon2bi 2650 . . 3  |-  ( 0  =  0  ->  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )
84, 7ax-mp 8 . 2  |-  -.  (
1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )
9 ndmovg 6230 . 2  |-  ( ( dom  /  =  ( CC  X.  ( CC 
\  { 0 } ) )  /\  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )  -> 
( 1  /  0
)  =  (/) )
103, 8, 9mp2an 654 1  |-  ( 1  /  0 )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 359    = wceq 1652    e. wcel 1725    =/= wne 2599    \ cdif 3317   (/)c0 3628   {csn 3814    X. cxp 4876   dom cdm 4878  (class class class)co 6081   iota_crio 6542   CCcc 8988   0cc0 8990   1c1 8991    x. cmul 8995    / cdiv 9677
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-csb 3252  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-iun 4095  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-fv 5462  df-ov 6084  df-oprab 6085  df-mpt2 6086  df-1st 6349  df-2nd 6350  df-riota 6549  df-div 9678
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