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Theorem en1b 7175
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.)
Assertion
Ref Expression
en1b  |-  ( A 
~~  1o  <->  A  =  { U. A } )

Proof of Theorem en1b
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 en1 7174 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
2 id 20 . . . . 5  |-  ( A  =  { x }  ->  A  =  { x } )
3 unieq 4024 . . . . . . 7  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
4 vex 2959 . . . . . . . 8  |-  x  e. 
_V
54unisn 4031 . . . . . . 7  |-  U. {
x }  =  x
63, 5syl6eq 2484 . . . . . 6  |-  ( A  =  { x }  ->  U. A  =  x )
76sneqd 3827 . . . . 5  |-  ( A  =  { x }  ->  { U. A }  =  { x } )
82, 7eqtr4d 2471 . . . 4  |-  ( A  =  { x }  ->  A  =  { U. A } )
98exlimiv 1644 . . 3  |-  ( E. x  A  =  {
x }  ->  A  =  { U. A }
)
101, 9sylbi 188 . 2  |-  ( A 
~~  1o  ->  A  =  { U. A }
)
11 id 20 . . 3  |-  ( A  =  { U. A }  ->  A  =  { U. A } )
12 snex 4405 . . . . . 6  |-  { U. A }  e.  _V
1311, 12syl6eqel 2524 . . . . 5  |-  ( A  =  { U. A }  ->  A  e.  _V )
14 uniexg 4706 . . . . 5  |-  ( A  e.  _V  ->  U. A  e.  _V )
1513, 14syl 16 . . . 4  |-  ( A  =  { U. A }  ->  U. A  e.  _V )
16 ensn1g 7172 . . . 4  |-  ( U. A  e.  _V  ->  { U. A }  ~~  1o )
1715, 16syl 16 . . 3  |-  ( A  =  { U. A }  ->  { U. A }  ~~  1o )
1811, 17eqbrtrd 4232 . 2  |-  ( A  =  { U. A }  ->  A  ~~  1o )
1910, 18impbii 181 1  |-  ( A 
~~  1o  <->  A  =  { U. A } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725   _Vcvv 2956   {csn 3814   U.cuni 4015   class class class wbr 4212   1oc1o 6717    ~~ cen 7106
This theorem is referenced by:  sylow2alem2  15252  sylow2a  15253  frgpcyg  16854  ptcmplem3  18085  cnextfvval  18096  cnextcn  18098  minveclem4a  19331  isppw  20897  xrge0tsmsbi  24224  en1uniel  27357
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-13 1727  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-reu 2712  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-nul 3629  df-if 3740  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-id 4498  df-suc 4587  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-res 4890  df-ima 4891  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-1o 6724  df-en 7110
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