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Theorem 1div0apr 21793
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 9709 . . 3  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC ( y  x.  z )  =  x ) )
2 riotaex 6582 . . 3  |-  ( iota_ z  e.  CC ( y  x.  z )  =  x )  e.  _V
31, 2dmmpt2 6450 . 2  |-  dom  /  =  ( CC  X.  ( CC  \  { 0 } ) )
4 eqid 2442 . . 3  |-  0  =  0
5 eldifsni 3952 . . . . 5  |-  ( 0  e.  ( CC  \  { 0 } )  ->  0  =/=  0
)
65adantl 454 . . . 4  |-  ( ( 1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )  ->  0  =/=  0 )
76necon2bi 2656 . . 3  |-  ( 0  =  0  ->  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )
84, 7ax-mp 5 . 2  |-  -.  (
1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )
9 ndmovg 6259 . 2  |-  ( ( dom  /  =  ( CC  X.  ( CC 
\  { 0 } ) )  /\  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )  -> 
( 1  /  0
)  =  (/) )
103, 8, 9mp2an 655 1  |-  ( 1  /  0 )  =  (/)
Colors of variables: wff set class
Syntax hints:   -. wn 3    /\ wa 360    = wceq 1653    e. wcel 1727    =/= wne 2605    \ cdif 3303   (/)c0 3613   {csn 3838    X. cxp 4905   dom cdm 4907  (class class class)co 6110   iota_crio 6571   CCcc 9019   0cc0 9021   1c1 9022    x. cmul 9026    / cdiv 9708
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1668  ax-8 1689  ax-13 1729  ax-14 1731  ax-6 1746  ax-7 1751  ax-11 1763  ax-12 1953  ax-ext 2423  ax-sep 4355  ax-nul 4363  ax-pow 4406  ax-pr 4432  ax-un 4730
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2291  df-mo 2292  df-clab 2429  df-cleq 2435  df-clel 2438  df-nfc 2567  df-ne 2607  df-ral 2716  df-rex 2717  df-rab 2720  df-v 2964  df-sbc 3168  df-csb 3268  df-dif 3309  df-un 3311  df-in 3313  df-ss 3320  df-nul 3614  df-if 3764  df-sn 3844  df-pr 3845  df-op 3847  df-uni 4040  df-iun 4119  df-br 4238  df-opab 4292  df-mpt 4293  df-id 4527  df-xp 4913  df-rel 4914  df-cnv 4915  df-co 4916  df-dm 4917  df-rn 4918  df-res 4919  df-ima 4920  df-iota 5447  df-fun 5485  df-fn 5486  df-f 5487  df-fv 5491  df-ov 6113  df-oprab 6114  df-mpt2 6115  df-1st 6378  df-2nd 6379  df-riota 6578  df-div 9709
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