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Theorem card1 7855
Description: A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.)
Assertion
Ref Expression
card1  |-  ( (
card `  A )  =  1o  <->  E. x  A  =  { x } )
Distinct variable group:    x, A

Proof of Theorem card1
StepHypRef Expression
1 1onn 6882 . . . . . . . 8  |-  1o  e.  om
2 cardnn 7850 . . . . . . . 8  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
31, 2ax-mp 8 . . . . . . 7  |-  ( card `  1o )  =  1o
4 1n0 6739 . . . . . . 7  |-  1o  =/=  (/)
53, 4eqnetri 2618 . . . . . 6  |-  ( card `  1o )  =/=  (/)
6 carden2a 7853 . . . . . 6  |-  ( ( ( card `  1o )  =  ( card `  A )  /\  ( card `  1o )  =/=  (/) )  ->  1o  ~~  A )
75, 6mpan2 653 . . . . 5  |-  ( (
card `  1o )  =  ( card `  A
)  ->  1o  ~~  A
)
87eqcoms 2439 . . . 4  |-  ( (
card `  A )  =  ( card `  1o )  ->  1o  ~~  A
)
98ensymd 7158 . . 3  |-  ( (
card `  A )  =  ( card `  1o )  ->  A  ~~  1o )
10 carden2b 7854 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
119, 10impbii 181 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o )
123eqeq2i 2446 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
13 en1 7174 . 2  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
1411, 12, 133bitr3i 267 1  |-  ( (
card `  A )  =  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 177   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   {csn 3814   class class class wbr 4212   omcom 4845   ` cfv 5454   1oc1o 6717    ~~ cen 7106   cardccrd 7822
This theorem is referenced by:  cardsn  7856
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-pow 4377  ax-pr 4403  ax-un 4701
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3or 937  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-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-tp 3822  df-op 3823  df-uni 4016  df-int 4051  df-br 4213  df-opab 4267  df-mpt 4268  df-tr 4303  df-eprel 4494  df-id 4498  df-po 4503  df-so 4504  df-fr 4541  df-we 4543  df-ord 4584  df-on 4585  df-lim 4586  df-suc 4587  df-om 4846  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-er 6905  df-en 7110  df-dom 7111  df-sdom 7112  df-fin 7113  df-card 7826
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