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Theorem pr1eqbg 31991
Description: A (proper) pair is equal to another (maybe inproper) pair containing one element of the first pair if and only if the other element of the first pair is contained in the second pair. (Contributed by Alexander van der Vekens, 26-Jan-2018.)
Assertion
Ref Expression
pr1eqbg  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  ( A  =  C  <->  { A ,  B }  =  { B ,  C } ) )

Proof of Theorem pr1eqbg
StepHypRef Expression
1 eqid 2467 . . . . 5  |-  B  =  B
21biantru 505 . . . 4  |-  ( A  =  C  <->  ( A  =  C  /\  B  =  B ) )
32orbi2i 519 . . 3  |-  ( ( ( A  =  B  /\  B  =  C )  \/  A  =  C )  <->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) ) )
43a1i 11 . 2  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  ( (
( A  =  B  /\  B  =  C )  \/  A  =  C )  <->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) ) ) )
5 df-ne 2664 . . . . . 6  |-  ( A  =/=  B  <->  -.  A  =  B )
65biimpi 194 . . . . 5  |-  ( A  =/=  B  ->  -.  A  =  B )
76adantl 466 . . . 4  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  -.  A  =  B )
87intnanrd 915 . . 3  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  -.  ( A  =  B  /\  B  =  C )
)
9 biorf 405 . . 3  |-  ( -.  ( A  =  B  /\  B  =  C )  ->  ( A  =  C  <->  ( ( A  =  B  /\  B  =  C )  \/  A  =  C ) ) )
108, 9syl 16 . 2  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  ( A  =  C  <->  ( ( A  =  B  /\  B  =  C )  \/  A  =  C ) ) )
11 3simpa 993 . . . . 5  |-  ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X )  ->  ( A  e.  U  /\  B  e.  V
) )
12 3simpc 995 . . . . 5  |-  ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X )  ->  ( B  e.  V  /\  C  e.  X
) )
1311, 12jca 532 . . . 4  |-  ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X )  ->  ( ( A  e.  U  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  X )
) )
1413adantr 465 . . 3  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  ( ( A  e.  U  /\  B  e.  V )  /\  ( B  e.  V  /\  C  e.  X
) ) )
15 preq12bg 4205 . . 3  |-  ( ( ( A  e.  U  /\  B  e.  V
)  /\  ( B  e.  V  /\  C  e.  X ) )  -> 
( { A ,  B }  =  { B ,  C }  <->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B ) ) ) )
1614, 15syl 16 . 2  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  ( { A ,  B }  =  { B ,  C } 
<->  ( ( A  =  B  /\  B  =  C )  \/  ( A  =  C  /\  B  =  B )
) ) )
174, 10, 163bitr4d 285 1  |-  ( ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X
)  /\  A  =/=  B )  ->  ( A  =  C  <->  { A ,  B }  =  { B ,  C } ) )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 184    \/ wo 368    /\ wa 369    /\ w3a 973    = wceq 1379    e. wcel 1767    =/= wne 2662   {cpr 4029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-v 3115  df-un 3481  df-sn 4028  df-pr 4030
This theorem is referenced by:  pr1nebg  31992
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