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Theorem card1 7617
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 6653 . . . . . . . 8  |-  1o  e.  om
2 cardnn 7612 . . . . . . . 8  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
31, 2ax-mp 8 . . . . . . 7  |-  ( card `  1o )  =  1o
4 1n0 6510 . . . . . . 7  |-  1o  =/=  (/)
53, 4eqnetri 2476 . . . . . 6  |-  ( card `  1o )  =/=  (/)
6 carden2a 7615 . . . . . 6  |-  ( ( ( card `  1o )  =  ( card `  A )  /\  ( card `  1o )  =/=  (/) )  ->  1o  ~~  A )
75, 6mpan2 652 . . . . 5  |-  ( (
card `  1o )  =  ( card `  A
)  ->  1o  ~~  A
)
87eqcoms 2299 . . . 4  |-  ( (
card `  A )  =  ( card `  1o )  ->  1o  ~~  A
)
9 ensym 6926 . . . 4  |-  ( 1o 
~~  A  ->  A  ~~  1o )
108, 9syl 15 . . 3  |-  ( (
card `  A )  =  ( card `  1o )  ->  A  ~~  1o )
11 carden2b 7616 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
1210, 11impbii 180 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o )
133eqeq2i 2306 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
14 en1 6944 . 2  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
1512, 13, 143bitr3i 266 1  |-  ( (
card `  A )  =  1o  <->  E. x  A  =  { x } )
Colors of variables: wff set class
Syntax hints:    <-> wb 176   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   {csn 3653   class class class wbr 4039   omcom 4672   ` cfv 5271   1oc1o 6488    ~~ cen 6876   cardccrd 7584
This theorem is referenced by:  cardsn  7618
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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