Theorem fieq0 7300

Description: A set is empty iff the class of all the finite intersections of that set is empty. (Contributed by FL, 27-Apr-2008.) (Revised by Mario Carneiro, 24-Nov-2013.)

Assertion

Ref	Expression
fieq0	|- ( A e. V -> ( A = (/) <-> ( fi ` A ) = (/) ) )

Proof of Theorem fieq0

Step	Hyp	Ref	Expression
1		fveq2 5690	. . 3 |- ( A = (/) -> ( fi ` A ) = ( fi ` (/) ) )
2		fi0 7299	. . 3 |- ( fi ` (/) ) = (/)
3	1, 2	eqtrdi 2287	. 2 |- ( A = (/) -> ( fi ` A ) = (/) )
4		ssfii 7298	. . . 4 |- ( A e. V -> A C_ ( fi ` A ) )
5		sseq0 3564	. . . 4 |- ( ( A C_ ( fi ` A ) /\ ( fi ` A ) = (/) ) -> A = (/) )
6	4, 5	sylan 283	. . 3 |- ( ( A e. V /\ ( fi ` A ) = (/) ) -> A = (/) )
7	6	ex 115	. 2 |- ( A e. V -> ( ( fi ` A ) = (/) -> A = (/) ) )
8	3, 7	impbid2 143	1 |- ( A e. V -> ( A = (/) <-> ( fi ` A ) = (/) ) )

Syntax hints:   -> wi 4   <-> wb 105   = wceq 1402   e. wcel 2209   C_ wss 3220   (/)c0 3520   ` cfv 5372   ficfi 7292

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-13 2211  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-nul 4254  ax-pow 4306  ax-pr 4341  ax-un 4573  ax-iinf 4730

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-int 3966  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-suc 4511  df-iom 4733  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-1o 6677  df-er 6797  df-en 7013  df-fin 7015  df-fi 7293

This theorem is referenced by: (None)