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Theorem apirr 8233
Description: Apartness is irreflexive. (Contributed by Jim Kingdon, 16-Feb-2020.)
Assertion
Ref Expression
apirr  |-  ( A  e.  CC  ->  -.  A #  A )

Proof of Theorem apirr
Dummy variables  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cnre 7634 . 2  |-  ( A  e.  CC  ->  E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y ) ) )
2 reapirr 8205 . . . . . . . . . 10  |-  ( x  e.  RR  ->  -.  x #  x )
3 apreap 8215 . . . . . . . . . . 11  |-  ( ( x  e.  RR  /\  x  e.  RR )  ->  ( x #  x  <->  x #  x )
)
43anidms 392 . . . . . . . . . 10  |-  ( x  e.  RR  ->  (
x #  x  <->  x #  x )
)
52, 4mtbird 639 . . . . . . . . 9  |-  ( x  e.  RR  ->  -.  x #  x )
6 reapirr 8205 . . . . . . . . . 10  |-  ( y  e.  RR  ->  -.  y #  y )
7 apreap 8215 . . . . . . . . . . 11  |-  ( ( y  e.  RR  /\  y  e.  RR )  ->  ( y #  y  <->  y #  y )
)
87anidms 392 . . . . . . . . . 10  |-  ( y  e.  RR  ->  (
y #  y  <->  y #  y )
)
96, 8mtbird 639 . . . . . . . . 9  |-  ( y  e.  RR  ->  -.  y #  y )
105, 9anim12i 334 . . . . . . . 8  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( -.  x #  x  /\  -.  y #  y ) )
11 ioran 710 . . . . . . . 8  |-  ( -.  ( x #  x  \/  y #  y )  <->  ( -.  x #  x  /\  -.  y #  y ) )
1210, 11sylibr 133 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  -.  ( x #  x  \/  y #  y )
)
13 apreim 8231 . . . . . . . 8  |-  ( ( ( x  e.  RR  /\  y  e.  RR )  /\  ( x  e.  RR  /\  y  e.  RR ) )  -> 
( ( x  +  ( _i  x.  y
) ) #  ( x  +  ( _i  x.  y ) )  <->  ( x #  x  \/  y #  y
) ) )
1413anidms 392 . . . . . . 7  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  ( ( x  +  ( _i  x.  y
) ) #  ( x  +  ( _i  x.  y ) )  <->  ( x #  x  \/  y #  y
) ) )
1512, 14mtbird 639 . . . . . 6  |-  ( ( x  e.  RR  /\  y  e.  RR )  ->  -.  ( x  +  ( _i  x.  y
) ) #  ( x  +  ( _i  x.  y ) ) )
1615ad2antlr 476 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  ->  -.  ( x  +  (
_i  x.  y )
) #  ( x  +  ( _i  x.  y
) ) )
17 id 19 . . . . . . . 8  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  A  =  ( x  +  ( _i  x.  y
) ) )
1817, 17breq12d 3888 . . . . . . 7  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  ( A #  A  <->  ( x  +  ( _i  x.  y
) ) #  ( x  +  ( _i  x.  y ) ) ) )
1918notbid 633 . . . . . 6  |-  ( A  =  ( x  +  ( _i  x.  y
) )  ->  ( -.  A #  A  <->  -.  (
x  +  ( _i  x.  y ) ) #  ( x  +  ( _i  x.  y ) ) ) )
2019adantl 273 . . . . 5  |-  ( ( ( A  e.  CC  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  ->  ( -.  A #  A  <->  -.  (
x  +  ( _i  x.  y ) ) #  ( x  +  ( _i  x.  y ) ) ) )
2116, 20mpbird 166 . . . 4  |-  ( ( ( A  e.  CC  /\  ( x  e.  RR  /\  y  e.  RR ) )  /\  A  =  ( x  +  ( _i  x.  y ) ) )  ->  -.  A #  A )
2221ex 114 . . 3  |-  ( ( A  e.  CC  /\  ( x  e.  RR  /\  y  e.  RR ) )  ->  ( A  =  ( x  +  ( _i  x.  y
) )  ->  -.  A #  A ) )
2322rexlimdvva 2516 . 2  |-  ( A  e.  CC  ->  ( E. x  e.  RR  E. y  e.  RR  A  =  ( x  +  ( _i  x.  y
) )  ->  -.  A #  A ) )
241, 23mpd 13 1  |-  ( A  e.  CC  ->  -.  A #  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 670    = wceq 1299    e. wcel 1448   E.wrex 2376   class class class wbr 3875  (class class class)co 5706   CCcc 7498   RRcr 7499   _ici 7502    + caddc 7503    x. cmul 7505   # creap 8202   # cap 8209
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-13 1459  ax-14 1460  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082  ax-sep 3986  ax-pow 4038  ax-pr 4069  ax-un 4293  ax-setind 4390  ax-cnex 7586  ax-resscn 7587  ax-1cn 7588  ax-1re 7589  ax-icn 7590  ax-addcl 7591  ax-addrcl 7592  ax-mulcl 7593  ax-mulrcl 7594  ax-addcom 7595  ax-mulcom 7596  ax-addass 7597  ax-mulass 7598  ax-distr 7599  ax-i2m1 7600  ax-0lt1 7601  ax-1rid 7602  ax-0id 7603  ax-rnegex 7604  ax-precex 7605  ax-cnre 7606  ax-pre-ltirr 7607  ax-pre-lttrn 7609  ax-pre-apti 7610  ax-pre-ltadd 7611  ax-pre-mulgt0 7612
This theorem depends on definitions:  df-bi 116  df-3an 932  df-tru 1302  df-fal 1305  df-nf 1405  df-sb 1704  df-eu 1963  df-mo 1964  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-ral 2380  df-rex 2381  df-reu 2382  df-rab 2384  df-v 2643  df-sbc 2863  df-dif 3023  df-un 3025  df-in 3027  df-ss 3034  df-pw 3459  df-sn 3480  df-pr 3481  df-op 3483  df-uni 3684  df-br 3876  df-opab 3930  df-id 4153  df-xp 4483  df-rel 4484  df-cnv 4485  df-co 4486  df-dm 4487  df-iota 5024  df-fun 5061  df-fv 5067  df-riota 5662  df-ov 5709  df-oprab 5710  df-mpo 5711  df-pnf 7674  df-mnf 7675  df-ltxr 7677  df-sub 7806  df-neg 7807  df-reap 8203  df-ap 8210
This theorem is referenced by:  mulap0r  8243  eirr  11280
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