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Theorem card1 8399
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 7337 . . . . . . . 8  |-  1o  e.  om
2 cardnn 8394 . . . . . . . 8  |-  ( 1o  e.  om  ->  ( card `  1o )  =  1o )
31, 2ax-mp 5 . . . . . . 7  |-  ( card `  1o )  =  1o
4 1n0 7194 . . . . . . 7  |-  1o  =/=  (/)
53, 4eqnetri 2693 . . . . . 6  |-  ( card `  1o )  =/=  (/)
6 carden2a 8397 . . . . . 6  |-  ( ( ( card `  1o )  =  ( card `  A )  /\  ( card `  1o )  =/=  (/) )  ->  1o  ~~  A )
75, 6mpan2 676 . . . . 5  |-  ( (
card `  1o )  =  ( card `  A
)  ->  1o  ~~  A
)
87eqcoms 2458 . . . 4  |-  ( (
card `  A )  =  ( card `  1o )  ->  1o  ~~  A
)
98ensymd 7617 . . 3  |-  ( (
card `  A )  =  ( card `  1o )  ->  A  ~~  1o )
10 carden2b 8398 . . 3  |-  ( A 
~~  1o  ->  ( card `  A )  =  (
card `  1o )
)
119, 10impbii 191 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  A  ~~  1o )
123eqeq2i 2462 . 2  |-  ( (
card `  A )  =  ( card `  1o ) 
<->  ( card `  A
)  =  1o )
13 en1 7633 . 2  |-  ( A 
~~  1o  <->  E. x  A  =  { x } )
1411, 12, 133bitr3i 279 1  |-  ( (
card `  A )  =  1o  <->  E. x  A  =  { x } )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 188    = wceq 1443   E.wex 1662    e. wcel 1886    =/= wne 2621   (/)c0 3730   {csn 3967   class class class wbr 4401   ` cfv 5581   omcom 6689   1oc1o 7172    ~~ cen 7563   cardccrd 8366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1668  ax-4 1681  ax-5 1757  ax-6 1804  ax-7 1850  ax-8 1888  ax-9 1895  ax-10 1914  ax-11 1919  ax-12 1932  ax-13 2090  ax-ext 2430  ax-sep 4524  ax-nul 4533  ax-pow 4580  ax-pr 4638  ax-un 6580
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 985  df-3an 986  df-tru 1446  df-ex 1663  df-nf 1667  df-sb 1797  df-eu 2302  df-mo 2303  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2580  df-ne 2623  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3046  df-sbc 3267  df-dif 3406  df-un 3408  df-in 3410  df-ss 3417  df-pss 3419  df-nul 3731  df-if 3881  df-pw 3952  df-sn 3968  df-pr 3970  df-tp 3972  df-op 3974  df-uni 4198  df-int 4234  df-br 4402  df-opab 4461  df-mpt 4462  df-tr 4497  df-eprel 4744  df-id 4748  df-po 4754  df-so 4755  df-fr 4792  df-we 4794  df-xp 4839  df-rel 4840  df-cnv 4841  df-co 4842  df-dm 4843  df-rn 4844  df-res 4845  df-ima 4846  df-ord 5425  df-on 5426  df-lim 5427  df-suc 5428  df-iota 5545  df-fun 5583  df-fn 5584  df-f 5585  df-f1 5586  df-fo 5587  df-f1o 5588  df-fv 5589  df-om 6690  df-1o 7179  df-er 7360  df-en 7567  df-dom 7568  df-sdom 7569  df-fin 7570  df-card 8370
This theorem is referenced by:  cardsn  8400
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