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Theorem card1 8381
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 7325 . . . . . . . 8  |-  1o  e.  om
2 cardnn 8376 . . . . . . . 8  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
31, 2ax-mp 5 . . . . . . 7  |-  ( card `  1o )  =  1o
4 1n0 7182 . . . . . . 7  |-  1o  =/=  (/)
53, 4eqnetri 2699 . . . . . 6  |-  ( card `  1o )  =/=  (/)
6 carden2a 8379 . . . . . 6  |-  ( ( ( card `  1o )  =  ( card `  A )  /\  ( card `  1o )  =/=  (/) )  ->  1o  ~~  A )
75, 6mpan2 669 . . . . 5  |-  ( (
card `  1o )  =  ( card `  A
)  ->  1o  ~~  A
)
87eqcoms 2414 . . . 4  |-  ( (
card `  A )  =  ( card `  1o )  ->  1o  ~~  A
)
98ensymd 7604 . . 3  |-  ( (
card `  A )  =  ( card `  1o )  ->  A  ~~  1o )
10 carden2b 8380 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
119, 10impbii 187 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o )
123eqeq2i 2420 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
13 en1 7620 . 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 1405   E.wex 1633    e. wcel 1842    =/= wne 2598   (/)c0 3738   {csn 3972   class class class wbr 4395   ` cfv 5569   omcom 6683   1oc1o 7160    ~~ cen 7551   cardccrd 8348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4517  ax-nul 4525  ax-pow 4572  ax-pr 4630  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3or 975  df-3an 976  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-eu 2242  df-mo 2243  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2759  df-rex 2760  df-reu 2761  df-rab 2763  df-v 3061  df-sbc 3278  df-dif 3417  df-un 3419  df-in 3421  df-ss 3428  df-pss 3430  df-nul 3739  df-if 3886  df-pw 3957  df-sn 3973  df-pr 3975  df-tp 3977  df-op 3979  df-uni 4192  df-int 4228  df-br 4396  df-opab 4454  df-mpt 4455  df-tr 4490  df-eprel 4734  df-id 4738  df-po 4744  df-so 4745  df-fr 4782  df-we 4784  df-xp 4829  df-rel 4830  df-cnv 4831  df-co 4832  df-dm 4833  df-rn 4834  df-res 4835  df-ima 4836  df-ord 5413  df-on 5414  df-lim 5415  df-suc 5416  df-iota 5533  df-fun 5571  df-fn 5572  df-f 5573  df-f1 5574  df-fo 5575  df-f1o 5576  df-fv 5577  df-om 6684  df-1o 7167  df-er 7348  df-en 7555  df-dom 7556  df-sdom 7557  df-fin 7558  df-card 8352
This theorem is referenced by:  cardsn  8382
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