Theorem card1 8338

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 7278	. . . . . . . 8 |- 1o e. om
2		cardnn 8333	. . . . . . . 8 |- ( 1o e. om -> ( card ` 1o ) = 1o )
3	1, 2	ax-mp 5	. . . . . . 7 |- ( card ` 1o ) = 1o
4		1n0 7135	. . . . . . 7 |- 1o =/= (/)
5	3, 4	eqnetri 2756	. . . . . 6 |- ( card ` 1o ) =/= (/)
6		carden2a 8336	. . . . . 6 |- ( ( ( card ` 1o ) = ( card ` A ) /\ ( card ` 1o ) =/= (/) ) -> 1o ~~ A )
7	5, 6	mpan2 671	. . . . 5 |- ( ( card ` 1o ) = ( card ` A ) -> 1o ~~ A )
8	7	eqcoms 2472	. . . 4 |- ( ( card ` A ) = ( card ` 1o ) -> 1o ~~ A )
9	8	ensymd 7556	. . 3 |- ( ( card ` A ) = ( card ` 1o ) -> A ~~ 1o )
10		carden2b 8337	. . 3 |- ( A ~~ 1o -> ( card ` A ) = ( card ` 1o ) )
11	9, 10	impbii 188	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o )
12	3	eqeq2i 2478	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> ( card ` A ) = 1o )
13		en1 7572	. 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 1374   E.wex 1591   e. wcel 1762   =/= wne 2655   (/)c0 3778   {csn 4020   class class class wbr 4440   ` cfv 5579   omcom 6671   1oc1o 7113   ~~ cen 7503   cardccrd 8305

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1596  ax-4 1607  ax-5 1675  ax-6 1714  ax-7 1734  ax-8 1764  ax-9 1766  ax-10 1781  ax-11 1786  ax-12 1798  ax-13 1961  ax-ext 2438  ax-sep 4561  ax-nul 4569  ax-pow 4618  ax-pr 4679  ax-un 6567

This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 969  df-3an 970  df-tru 1377  df-ex 1592  df-nf 1595  df-sb 1707  df-eu 2272  df-mo 2273  df-clab 2446  df-cleq 2452  df-clel 2455  df-nfc 2610  df-ne 2657  df-ral 2812  df-rex 2813  df-reu 2814  df-rab 2816  df-v 3108  df-sbc 3325  df-dif 3472  df-un 3474  df-in 3476  df-ss 3483  df-pss 3485  df-nul 3779  df-if 3933  df-pw 4005  df-sn 4021  df-pr 4023  df-tp 4025  df-op 4027  df-uni 4239  df-int 4276  df-br 4441  df-opab 4499  df-mpt 4500  df-tr 4534  df-eprel 4784  df-id 4788  df-po 4793  df-so 4794  df-fr 4831  df-we 4833  df-ord 4874  df-on 4875  df-lim 4876  df-suc 4877  df-xp 4998  df-rel 4999  df-cnv 5000  df-co 5001  df-dm 5002  df-rn 5003  df-res 5004  df-ima 5005  df-iota 5542  df-fun 5581  df-fn 5582  df-f 5583  df-f1 5584  df-fo 5585  df-f1o 5586  df-fv 5587  df-om 6672  df-1o 7120  df-er 7301  df-en 7507  df-dom 7508  df-sdom 7509  df-fin 7510  df-card 8309

This theorem is referenced by:  cardsn  8339