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Theorem pr1eqbg 30270
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 2454 . . . . 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 2650 . . . . . 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 908 . . 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 985 . . . . 5  |-  ( ( A  e.  U  /\  B  e.  V  /\  C  e.  X )  ->  ( A  e.  U  /\  B  e.  V
) )
12 3simpc 987 . . . . 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 4160 . . 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 965    = wceq 1370    e. wcel 1758    =/= wne 2648   {cpr 3988
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-v 3080  df-un 3442  df-sn 3987  df-pr 3989
This theorem is referenced by:  pr1nebg  30271
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