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

Step	Hyp	Ref	Expression
1		1onn 7337	. . . . . . . 8 |- 1o e. om
2		cardnn 8394	. . . . . . . 8 |- ( 1o e. om -> ( card ` 1o ) = 1o )
3	1, 2	ax-mp 5	. . . . . . 7 |- ( card ` 1o ) = 1o
4		1n0 7194	. . . . . . 7 |- 1o =/= (/)
5	3, 4	eqnetri 2693	. . . . . 6 |- ( card ` 1o ) =/= (/)
6		carden2a 8397	. . . . . 6 |- ( ( ( card ` 1o ) = ( card ` A ) /\ ( card ` 1o ) =/= (/) ) -> 1o ~~ A )
7	5, 6	mpan2 676	. . . . 5 |- ( ( card ` 1o ) = ( card ` A ) -> 1o ~~ A )
8	7	eqcoms 2458	. . . 4 |- ( ( card ` A ) = ( card ` 1o ) -> 1o ~~ A )
9	8	ensymd 7617	. . 3 |- ( ( card ` A ) = ( card ` 1o ) -> A ~~ 1o )
10		carden2b 8398	. . 3 |- ( A ~~ 1o -> ( card ` A ) = ( card ` 1o ) )
11	9, 10	impbii 191	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> A ~~ 1o )
12	3	eqeq2i 2462	. 2 |- ( ( card ` A ) = ( card ` 1o ) <-> ( card ` A ) = 1o )
13		en1 7633	. 2 |- ( A ~~ 1o <-> E. x A = { x } )
14	11, 12, 13	3bitr3i 279	1 |- ( ( card ` A ) = 1o <-> E. x A = { x } )

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