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

Step	Hyp	Ref	Expression
1		1onn 7325	. . . . . . . 8 |- 1o e. om
2		cardnn 8376	. . . . . . . 8 |- ( 1o e. om -> ( card ` 1o ) = 1o )
3	1, 2	ax-mp 5	. . . . . . 7 |- ( card ` 1o ) = 1o
4		1n0 7182	. . . . . . 7 |- 1o =/= (/)
5	3, 4	eqnetri 2699	. . . . . 6 |- ( card ` 1o ) =/= (/)
6		carden2a 8379	. . . . . 6 |- ( ( ( card ` 1o ) = ( card ` A ) /\ ( card ` 1o ) =/= (/) ) -> 1o ~~ A )
7	5, 6	mpan2 669	. . . . 5 |- ( ( card ` 1o ) = ( card ` A ) -> 1o ~~ A )
8	7	eqcoms 2414	. . . 4 |- ( ( card ` A ) = ( card ` 1o ) -> 1o ~~ A )
9	8	ensymd 7604	. . 3 |- ( ( card ` A ) = ( card ` 1o ) -> A ~~ 1o )
10		carden2b 8380	. . 3 |- ( A ~~ 1o -> ( card ` A ) = ( card ` 1o ) )
11	9, 10	impbii 187	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o )
12	3	eqeq2i 2420	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> ( card ` A ) = 1o )
13		en1 7620	. 2 |- ( A ~~ 1o <-> E. x A = { x } )
14	11, 12, 13	3bitr3i 275	1 |- ( ( card ` A ) = 1o <-> E. x A = { x } )

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