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Theorem 1div0apr 21723
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 9642 . . 3  |-  /  =  ( x  e.  CC ,  y  e.  ( CC  \  { 0 } )  |->  ( iota_ z  e.  CC ( y  x.  z )  =  x ) )
2 riotaex 6520 . . 3  |-  ( iota_ z  e.  CC ( y  x.  z )  =  x )  e.  _V
31, 2dmmpt2 6388 . 2  |-  dom  /  =  ( CC  X.  ( CC  \  { 0 } ) )
4 eqid 2412 . . 3  |-  0  =  0
5 eldifsni 3896 . . . . 5  |-  ( 0  e.  ( CC  \  { 0 } )  ->  0  =/=  0
)
65adantl 453 . . . 4  |-  ( ( 1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )  ->  0  =/=  0 )
76necon2bi 2621 . . 3  |-  ( 0  =  0  ->  -.  ( 1  e.  CC  /\  0  e.  ( CC 
\  { 0 } ) ) )
84, 7ax-mp 8 . 2  |-  -.  (
1  e.  CC  /\  0  e.  ( CC  \  { 0 } ) )
9 ndmovg 6197 . 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 1649    e. wcel 1721    =/= wne 2575    \ cdif 3285   (/)c0 3596   {csn 3782    X. cxp 4843   dom cdm 4845  (class class class)co 6048   iota_crio 6509   CCcc 8952   0cc0 8954   1c1 8955    x. cmul 8959    / cdiv 9641
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-csb 3220  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-iun 4063  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-1st 6316  df-2nd 6317  df-riota 6516  df-div 9642
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