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Theorem reuen1 7481
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 4049 . 2  |-  ( E! x  e.  A  ph  <->  E. y { x  e.  A  |  ph }  =  { y } )
2 en1 7479 . 2  |-  ( { x  e.  A  |  ph }  ~~  1o  <->  E. y { x  e.  A  |  ph }  =  {
y } )
31, 2bitr4i 252 1  |-  ( E! x  e.  A  ph  <->  { x  e.  A  |  ph }  ~~  1o )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370   E.wex 1587   E!wreu 2797   {crab 2799   {csn 3978   class class class wbr 4393   1oc1o 7016    ~~ cen 7410
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-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430  ax-sep 4514  ax-nul 4522  ax-pr 4632  ax-un 6475
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-eu 2264  df-mo 2265  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ne 2646  df-ral 2800  df-rex 2801  df-reu 2802  df-rab 2804  df-v 3073  df-sbc 3288  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-opab 4452  df-id 4737  df-suc 4826  df-xp 4947  df-rel 4948  df-cnv 4949  df-co 4950  df-dm 4951  df-rn 4952  df-res 4953  df-ima 4954  df-iota 5482  df-fun 5521  df-fn 5522  df-f 5523  df-f1 5524  df-fo 5525  df-f1o 5526  df-fv 5527  df-1o 7023  df-en 7414
This theorem is referenced by:  euen1  7482  isppw  22578
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