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

Step	Hyp	Ref	Expression
1		1onn 7189	. . . . . . . 8 |- 1o e. om
2		cardnn 8245	. . . . . . . 8 |- ( 1o e. om -> ( card ` 1o ) = 1o )
3	1, 2	ax-mp 5	. . . . . . 7 |- ( card ` 1o ) = 1o
4		1n0 7046	. . . . . . 7 |- 1o =/= (/)
5	3, 4	eqnetri 2748	. . . . . 6 |- ( card ` 1o ) =/= (/)
6		carden2a 8248	. . . . . 6 |- ( ( ( card ` 1o ) = ( card ` A ) /\ ( card ` 1o ) =/= (/) ) -> 1o ~~ A )
7	5, 6	mpan2 671	. . . . 5 |- ( ( card ` 1o ) = ( card ` A ) -> 1o ~~ A )
8	7	eqcoms 2466	. . . 4 |- ( ( card ` A ) = ( card ` 1o ) -> 1o ~~ A )
9	8	ensymd 7471	. . 3 |- ( ( card ` A ) = ( card ` 1o ) -> A ~~ 1o )
10		carden2b 8249	. . 3 |- ( A ~~ 1o -> ( card ` A ) = ( card ` 1o ) )
11	9, 10	impbii 188	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o )
12	3	eqeq2i 2472	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> ( card ` A ) = 1o )
13		en1 7487	. 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 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