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Theorem reuen1 6973
Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
reuen1  |-  ( E! x  e.  A  ph  <->  { x  e.  A  |  ph }  ~~  1o )
Distinct variable group:    x, A
Allowed substitution hint:    ph( x)

Proof of Theorem reuen1
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 reusn 3734 . 2  |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
2 en1 6971 . 2  |-  ( { x  e.  A  |  ph }  ~~  1o  <->  E. y { x  e.  A  |  ph }  =  {
y } )
31, 2bitr4i 243 1  |-  ( E! x  e.  A  ph  <->  { x  e.  A  |  ph }  ~~  1o )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1532    = wceq 1633   E!wreu 2579   {crab 2581   {csn 3674   class class class wbr 4060   1oc1o 6514    ~~ cen 6903
This theorem is referenced by:  euen1  6974  isppw  20405
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1537  ax-5 1548  ax-17 1607  ax-9 1645  ax-8 1666  ax-13 1703  ax-14 1705  ax-6 1720  ax-7 1725  ax-11 1732  ax-12 1897  ax-ext 2297  ax-sep 4178  ax-nul 4186  ax-pr 4251  ax-un 4549
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1533  df-nf 1536  df-sb 1640  df-eu 2180  df-mo 2181  df-clab 2303  df-cleq 2309  df-clel 2312  df-nfc 2441  df-ne 2481  df-ral 2582  df-rex 2583  df-reu 2584  df-rab 2586  df-v 2824  df-sbc 3026  df-dif 3189  df-un 3191  df-in 3193  df-ss 3200  df-nul 3490  df-if 3600  df-sn 3680  df-pr 3681  df-op 3683  df-uni 3865  df-br 4061  df-opab 4115  df-id 4346  df-suc 4435  df-xp 4732  df-rel 4733  df-cnv 4734  df-co 4735  df-dm 4736  df-rn 4737  df-res 4738  df-ima 4739  df-iota 5256  df-fun 5294  df-fn 5295  df-f 5296  df-f1 5297  df-fo 5298  df-f1o 5299  df-fv 5300  df-1o 6521  df-en 6907
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