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Theorem euen1b 6932
Description: Two ways to express " A has a unique element". (Contributed by Mario Carneiro, 9-Apr-2015.)
Assertion
Ref Expression
euen1b  |-  ( A 
~~  1o  <->  E! x  x  e.  A )
Distinct variable group:    x, A

Proof of Theorem euen1b
StepHypRef Expression
1 euen1 6931 . 2  |-  ( E! x  x  e.  A  <->  { x  |  x  e.  A }  ~~  1o )
2 abid2 2400 . . 3  |-  { x  |  x  e.  A }  =  A
32breq1i 4030 . 2  |-  ( { x  |  x  e.  A }  ~~  1o  <->  A 
~~  1o )
41, 3bitr2i 241 1  |-  ( A 
~~  1o  <->  E! x  x  e.  A )
Colors of variables: wff set class
Syntax hints:    <-> wb 176    e. wcel 1684   E!weu 2143   {cab 2269   class class class wbr 4023   1oc1o 6472    ~~ cen 6860
This theorem is referenced by:  hausflf2  17693  minveclem4a  18794  unexun  25569  f1otrspeq  27390
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-reu 2550  df-rab 2552  df-v 2790  df-sbc 2992  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-res 4701  df-ima 4702  df-iota 5219  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-fv 5263  df-1o 6479  df-en 6864
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