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Theorem elpr2elpr 39150
Description: For an element of an unordered pair which is a subset of a given set, there is another (maybe the same) element of the given set being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)
Assertion
Ref Expression
elpr2elpr  |-  ( ( X  e.  V  /\  Y  e.  V  /\  A  e.  { X ,  Y } )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b } )
Distinct variable groups:    A, b    V, b    X, b    Y, b

Proof of Theorem elpr2elpr
StepHypRef Expression
1 elpri 3976 . . . 4  |-  ( A  e.  { X ,  Y }  ->  ( A  =  X  \/  A  =  Y ) )
2 simprr 774 . . . . . . 7  |-  ( ( A  =  X  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  Y  e.  V )
3 preq2 4043 . . . . . . . . 9  |-  ( b  =  Y  ->  { A ,  b }  =  { A ,  Y }
)
43eqeq2d 2481 . . . . . . . 8  |-  ( b  =  Y  ->  ( { X ,  Y }  =  { A ,  b }  <->  { X ,  Y }  =  { A ,  Y } ) )
54adantl 473 . . . . . . 7  |-  ( ( ( A  =  X  /\  ( X  e.  V  /\  Y  e.  V ) )  /\  b  =  Y )  ->  ( { X ,  Y }  =  { A ,  b }  <->  { X ,  Y }  =  { A ,  Y } ) )
6 preq1 4042 . . . . . . . . 9  |-  ( X  =  A  ->  { X ,  Y }  =  { A ,  Y }
)
76eqcoms 2479 . . . . . . . 8  |-  ( A  =  X  ->  { X ,  Y }  =  { A ,  Y }
)
87adantr 472 . . . . . . 7  |-  ( ( A  =  X  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  { X ,  Y }  =  { A ,  Y }
)
92, 5, 8rspcedvd 3143 . . . . . 6  |-  ( ( A  =  X  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b }
)
109ex 441 . . . . 5  |-  ( A  =  X  ->  (
( X  e.  V  /\  Y  e.  V
)  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b }
) )
11 simprl 772 . . . . . . 7  |-  ( ( A  =  Y  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  X  e.  V )
12 preq2 4043 . . . . . . . . 9  |-  ( b  =  X  ->  { A ,  b }  =  { A ,  X }
)
1312eqeq2d 2481 . . . . . . . 8  |-  ( b  =  X  ->  ( { X ,  Y }  =  { A ,  b }  <->  { X ,  Y }  =  { A ,  X } ) )
1413adantl 473 . . . . . . 7  |-  ( ( ( A  =  Y  /\  ( X  e.  V  /\  Y  e.  V ) )  /\  b  =  X )  ->  ( { X ,  Y }  =  { A ,  b }  <->  { X ,  Y }  =  { A ,  X } ) )
15 preq2 4043 . . . . . . . . . 10  |-  ( Y  =  A  ->  { X ,  Y }  =  { X ,  A }
)
1615eqcoms 2479 . . . . . . . . 9  |-  ( A  =  Y  ->  { X ,  Y }  =  { X ,  A }
)
17 prcom 4041 . . . . . . . . 9  |-  { X ,  A }  =  { A ,  X }
1816, 17syl6eq 2521 . . . . . . . 8  |-  ( A  =  Y  ->  { X ,  Y }  =  { A ,  X }
)
1918adantr 472 . . . . . . 7  |-  ( ( A  =  Y  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  { X ,  Y }  =  { A ,  X }
)
2011, 14, 19rspcedvd 3143 . . . . . 6  |-  ( ( A  =  Y  /\  ( X  e.  V  /\  Y  e.  V
) )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b }
)
2120ex 441 . . . . 5  |-  ( A  =  Y  ->  (
( X  e.  V  /\  Y  e.  V
)  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b }
) )
2210, 21jaoi 386 . . . 4  |-  ( ( A  =  X  \/  A  =  Y )  ->  ( ( X  e.  V  /\  Y  e.  V )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b }
) )
231, 22syl 17 . . 3  |-  ( A  e.  { X ,  Y }  ->  ( ( X  e.  V  /\  Y  e.  V )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b } ) )
2423com12 31 . 2  |-  ( ( X  e.  V  /\  Y  e.  V )  ->  ( A  e.  { X ,  Y }  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b } ) )
25243impia 1228 1  |-  ( ( X  e.  V  /\  Y  e.  V  /\  A  e.  { X ,  Y } )  ->  E. b  e.  V  { X ,  Y }  =  { A ,  b } )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    <-> wb 189    \/ wo 375    /\ wa 376    /\ w3a 1007    = wceq 1452    e. wcel 1904   E.wrex 2757   {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-3an 1009  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-ral 2761  df-rex 2762  df-v 3033  df-un 3395  df-sn 3960  df-pr 3962
This theorem is referenced by:  upgredg2vtx  39392
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