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Theorem euen1 6947
Description: Two ways to express "exactly one". (Contributed by Stefan O'Rear, 28-Oct-2014.)
Assertion
Ref Expression
euen1  |-  ( E! x ph  <->  { x  |  ph }  ~~  1o )

Proof of Theorem euen1
StepHypRef Expression
1 reuen1 6946 . 2  |-  ( E! x  e.  _V  ph  <->  { x  e.  _V  |  ph }  ~~  1o )
2 reuv 2816 . 2  |-  ( E! x  e.  _V  ph  <->  E! x ph )
3 rabab 2818 . . 3  |-  { x  e.  _V  |  ph }  =  { x  |  ph }
43breq1i 4046 . 2  |-  ( { x  e.  _V  |  ph }  ~~  1o  <->  { x  |  ph }  ~~  1o )
51, 2, 43bitr3i 266 1  |-  ( E! x ph  <->  { x  |  ph }  ~~  1o )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E!weu 2156   {cab 2282   E!wreu 2558   {crab 2560   _Vcvv 2801   class class class wbr 4039   1oc1o 6488    ~~ cen 6876
This theorem is referenced by:  euen1b  6948  modom  7079
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-13 1698  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-reu 2563  df-rab 2565  df-v 2803  df-sbc 3005  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-uni 3844  df-br 4040  df-opab 4094  df-id 4325  df-suc 4414  df-xp 4711  df-rel 4712  df-cnv 4713  df-co 4714  df-dm 4715  df-rn 4716  df-res 4717  df-ima 4718  df-iota 5235  df-fun 5273  df-fn 5274  df-f 5275  df-f1 5276  df-fo 5277  df-f1o 5278  df-fv 5279  df-1o 6495  df-en 6880
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