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Theorem card1 8250
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 7189 . . . . . . . 8  |-  1o  e.  om
2 cardnn 8245 . . . . . . . 8  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
31, 2ax-mp 5 . . . . . . 7  |-  ( card `  1o )  =  1o
4 1n0 7046 . . . . . . 7  |-  1o  =/=  (/)
53, 4eqnetri 2748 . . . . . 6  |-  ( card `  1o )  =/=  (/)
6 carden2a 8248 . . . . . 6  |-  ( ( ( card `  1o )  =  ( card `  A )  /\  ( card `  1o )  =/=  (/) )  ->  1o  ~~  A )
75, 6mpan2 671 . . . . 5  |-  ( (
card `  1o )  =  ( card `  A
)  ->  1o  ~~  A
)
87eqcoms 2466 . . . 4  |-  ( (
card `  A )  =  ( card `  1o )  ->  1o  ~~  A
)
98ensymd 7471 . . 3  |-  ( (
card `  A )  =  ( card `  1o )  ->  A  ~~  1o )
10 carden2b 8249 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
119, 10impbii 188 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o )
123eqeq2i 2472 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
13 en1 7487 . 2  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
1411, 12, 133bitr3i 275 1  |-  ( (
card `  A )  =  1o  <->  E. x  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    = wceq 1370   E.wex 1587    e. wcel 1758    =/= wne 2648   (/)c0 3746   {csn 3986   class class class wbr 4401   ` cfv 5527   omcom 6587   1oc1o 7024    ~~ cen 7418   cardccrd 8217
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pow 4579  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-pss 3453  df-nul 3747  df-if 3901  df-pw 3971  df-sn 3987  df-pr 3989  df-tp 3991  df-op 3993  df-uni 4201  df-int 4238  df-br 4402  df-opab 4460  df-mpt 4461  df-tr 4495  df-eprel 4741  df-id 4745  df-po 4750  df-so 4751  df-fr 4788  df-we 4790  df-ord 4831  df-on 4832  df-lim 4833  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-om 6588  df-1o 7031  df-er 7212  df-en 7422  df-dom 7423  df-sdom 7424  df-fin 7425  df-card 8221
This theorem is referenced by:  cardsn  8251
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