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Theorem elprn1 37810
Description: A member of an unordered pair that is not the "first", must be the "second". (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Assertion
Ref Expression
elprn1  |-  ( ( A  e.  { B ,  C }  /\  A  =/=  B )  ->  A  =  C )

Proof of Theorem elprn1
StepHypRef Expression
1 neneq 2649 . . 3  |-  ( A  =/=  B  ->  -.  A  =  B )
21adantl 473 . 2  |-  ( ( A  e.  { B ,  C }  /\  A  =/=  B )  ->  -.  A  =  B )
3 elpri 3976 . . . 4  |-  ( A  e.  { B ,  C }  ->  ( A  =  B  \/  A  =  C ) )
43adantr 472 . . 3  |-  ( ( A  e.  { B ,  C }  /\  A  =/=  B )  ->  ( A  =  B  \/  A  =  C )
)
54ord 384 . 2  |-  ( ( A  e.  { B ,  C }  /\  A  =/=  B )  ->  ( -.  A  =  B  ->  A  =  C ) )
62, 5mpd 15 1  |-  ( ( A  e.  { B ,  C }  /\  A  =/=  B )  ->  A  =  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    \/ wo 375    /\ wa 376    = wceq 1452    e. wcel 1904    =/= wne 2641   {cpr 3961
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962
This theorem is referenced by:  fourierdlem70  38152  fourierdlem71  38153  fouriersw  38207  prsal  38291  sge0pr  38350
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