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Theorem en1b 7624
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 7623 . . 3  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
2 id 22 . . . . 5  |-  ( A  =  { x }  ->  A  =  { x } )
3 unieq 4176 . . . . . . 7  |-  ( A  =  { x }  ->  U. A  =  U. { x } )
4 vex 3016 . . . . . . . 8  |-  x  e. 
_V
54unisn 4183 . . . . . . 7  |-  U. {
x }  =  x
63, 5syl6eq 2502 . . . . . 6  |-  ( A  =  { x }  ->  U. A  =  x )
76sneqd 3948 . . . . 5  |-  ( A  =  { x }  ->  { U. A }  =  { x } )
82, 7eqtr4d 2489 . . . 4  |-  ( A  =  { x }  ->  A  =  { U. A } )
98exlimiv 1780 . . 3  |-  ( E. x  A  =  {
x }  ->  A  =  { U. A }
)
101, 9sylbi 200 . 2  |-  ( A 
~~  1o  ->  A  =  { U. A }
)
11 id 22 . . 3  |-  ( A  =  { U. A }  ->  A  =  { U. A } )
12 snex 4614 . . . . . 6  |-  { U. A }  e.  _V
1311, 12syl6eqel 2538 . . . . 5  |-  ( A  =  { U. A }  ->  A  e.  _V )
14 uniexg 6576 . . . . 5  |-  ( A  e.  _V  ->  U. A  e.  _V )
1513, 14syl 17 . . . 4  |-  ( A  =  { U. A }  ->  U. A  e.  _V )
16 ensn1g 7621 . . . 4  |-  ( U. A  e.  _V  ->  { U. A }  ~~  1o )
1715, 16syl 17 . . 3  |-  ( A  =  { U. A }  ->  { U. A }  ~~  1o )
1811, 17eqbrtrd 4395 . 2  |-  ( A  =  { U. A }  ->  A  ~~  1o )
1910, 18impbii 192 1  |-  ( A 
~~  1o  <->  A  =  { U. A } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 189    = wceq 1448   E.wex 1667    e. wcel 1891   _Vcvv 3013   {csn 3936   U.cuni 4168   class class class wbr 4374   1oc1o 7162    ~~ cen 7553
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1673  ax-4 1686  ax-5 1762  ax-6 1809  ax-7 1855  ax-8 1893  ax-9 1900  ax-10 1919  ax-11 1924  ax-12 1937  ax-13 2092  ax-ext 2432  ax-sep 4497  ax-nul 4506  ax-pr 4612  ax-un 6571
This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3an 988  df-tru 1451  df-ex 1668  df-nf 1672  df-sb 1802  df-eu 2304  df-mo 2305  df-clab 2439  df-cleq 2445  df-clel 2448  df-nfc 2582  df-ne 2624  df-ral 2742  df-rex 2743  df-reu 2744  df-rab 2746  df-v 3015  df-sbc 3236  df-dif 3375  df-un 3377  df-in 3379  df-ss 3386  df-nul 3700  df-if 3850  df-sn 3937  df-pr 3939  df-op 3943  df-uni 4169  df-br 4375  df-opab 4434  df-id 4727  df-xp 4818  df-rel 4819  df-cnv 4820  df-co 4821  df-dm 4822  df-rn 4823  df-res 4824  df-ima 4825  df-suc 5408  df-iota 5525  df-fun 5563  df-fn 5564  df-f 5565  df-f1 5566  df-fo 5567  df-f1o 5568  df-fv 5569  df-1o 7169  df-en 7557
This theorem is referenced by:  en1uniel  7628  sylow2alem2  17281  sylow2a  17282  frgpcyg  19155  ptcmplem3  21080  cnextfvval  21091  cnextcn  21093  minveclem4a  22383  minveclem4aOLD  22395  isppw  24053  xrge0tsmsbi  28556
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