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Theorem cardsn 8339
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 2460 . . 3  |-  { A }  =  { A }
2 sneq 4030 . . . . 5  |-  ( x  =  A  ->  { x }  =  { A } )
32eqeq2d 2474 . . . 4  |-  ( x  =  A  ->  ( { A }  =  {
x }  <->  { A }  =  { A } ) )
43spcegv 3192 . . 3  |-  ( A  e.  V  ->  ( { A }  =  { A }  ->  E. x { A }  =  {
x } ) )
51, 4mpi 17 . 2  |-  ( A  e.  V  ->  E. x { A }  =  {
x } )
6 card1 8338 . 2  |-  ( (
card `  { A } )  =  1o  <->  E. x { A }  =  { x } )
75, 6sylibr 212 1  |-  ( A  e.  V  ->  ( card `  { A }
)  =  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1374   E.wex 1591    e. wcel 1762   {csn 4020   ` cfv 5579   1oc1o 7113   cardccrd 8305
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309
This theorem is referenced by:  ackbij1lem14  8602  cfsuc  8626
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