Theorem metn0 18351

Description: A metric space is nonempty iff its base set is nonempty. (Contributed by NM, 4-Oct-2007.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
metn0	|- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )

Proof of Theorem metn0

Step	Hyp	Ref	Expression
1		metf 18321	. . . . 5 |- ( D e. ( Met ` X ) -> D : ( X X. X ) --> RR )
2		frel 5561	. . . . 5 |- ( D : ( X X. X ) --> RR -> Rel D )
3		reldm0 5054	. . . . 5 |- ( Rel D -> ( D = (/) <-> dom D = (/) ) )
4	1, 2, 3	3syl 19	. . . 4 |- ( D e. ( Met ` X ) -> ( D = (/) <-> dom D = (/) ) )
5		fdm 5562	. . . . . 6 |- ( D : ( X X. X ) --> RR -> dom D = ( X X. X ) )
6	1, 5	syl 16	. . . . 5 |- ( D e. ( Met ` X ) -> dom D = ( X X. X ) )
7	6	eqeq1d 2420	. . . 4 |- ( D e. ( Met ` X ) -> ( dom D = (/) <-> ( X X. X ) = (/) ) )
8	4, 7	bitrd 245	. . 3 |- ( D e. ( Met ` X ) -> ( D = (/) <-> ( X X. X ) = (/) ) )
9		xpeq0 5260	. . . 4 |- ( ( X X. X ) = (/) <-> ( X = (/) \/ X = (/) ) )
10		oridm 501	. . . 4 |- ( ( X = (/) \/ X = (/) ) <-> X = (/) )
11	9, 10	bitri 241	. . 3 |- ( ( X X. X ) = (/) <-> X = (/) )
12	8, 11	syl6bb 253	. 2 |- ( D e. ( Met ` X ) -> ( D = (/) <-> X = (/) ) )
13	12	necon3bid 2610	1 |- ( D e. ( Met ` X ) -> ( D =/= (/) <-> X =/= (/) ) )

Syntax hints:   -> wi 4   <-> wb 177   \/ wo 358   = wceq 1649   e. wcel 1721   =/= wne 2575   (/)c0 3596   X. cxp 4843   dom cdm 4845   Rel wrel 4850   -->wf 5417   ` cfv 5421   RRcr 8953   Metcme 16650

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2393  ax-sep 4298  ax-nul 4306  ax-pow 4345  ax-pr 4371  ax-un 4668  ax-cnex 9010  ax-resscn 9011

This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2266  df-mo 2267  df-clab 2399  df-cleq 2405  df-clel 2408  df-nfc 2537  df-ne 2577  df-ral 2679  df-rex 2680  df-rab 2683  df-v 2926  df-sbc 3130  df-dif 3291  df-un 3293  df-in 3295  df-ss 3302  df-nul 3597  df-if 3708  df-pw 3769  df-sn 3788  df-pr 3789  df-op 3791  df-uni 3984  df-br 4181  df-opab 4235  df-mpt 4236  df-id 4466  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-iota 5385  df-fun 5423  df-fn 5424  df-f 5425  df-fv 5429  df-ov 6051  df-oprab 6052  df-mpt2 6053  df-map 6987  df-met 16659