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Theorem cardsn 7618
Description: A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.)
Assertion
Ref Expression
cardsn  |-  ( A  e.  V  ->  ( card `  { A }
)  =  1o )

Proof of Theorem cardsn
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 eqid 2296 . . 3  |-  { A }  =  { A }
2 sneq 3664 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
32eqeq2d 2307 . . . 4  |-  ( x  =  A  ->  ( { A }  =  {
x }  <->  { A }  =  { A } ) )
43spcegv 2882 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { A }  ->  E. x { A }  =  {
x } ) )
51, 4mpi 16 . 2  |-  ( A  e.  V  ->  E. x { A }  =  {
x } )
6 card1 7617 . 2  |-  ( (
card `  { A } )  =  1o  <->  E. x { A }  =  { x } )
75, 6sylibr 203 1  |-  ( A  e.  V  ->  ( card `  { A }
)  =  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4   E.wex 1531    = wceq 1632    e. wcel 1696   {csn 3653   ` cfv 5271   1oc1o 6488   cardccrd 7584
This theorem is referenced by:  ackbij1lem14  7875  cfsuc  7899
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-pow 4204  ax-pr 4230  ax-un 4528
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  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-pss 3181  df-nul 3469  df-if 3579  df-pw 3640  df-sn 3659  df-pr 3660  df-tp 3661  df-op 3662  df-uni 3844  df-int 3879  df-br 4040  df-opab 4094  df-mpt 4095  df-tr 4130  df-eprel 4321  df-id 4325  df-po 4330  df-so 4331  df-fr 4368  df-we 4370  df-ord 4411  df-on 4412  df-lim 4413  df-suc 4414  df-om 4673  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-er 6676  df-en 6880  df-dom 6881  df-sdom 6882  df-fin 6883  df-card 7588
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