Theorem 0met 22171

Description: The empty metric. (Contributed by NM, 30-Aug-2006.) (Revised by Mario Carneiro, 14-Aug-2015.)

Assertion

Ref	Expression
0met	|- (/) e. ( Met ` (/) )

Proof of Theorem 0met

Dummy variables x y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		0ex 4790	. 2 |- (/) e. _V
2		f0 6086	. . 3 |- (/) : (/) --> RR
3		xp0 5552	. . . 4 |- ( (/) X. (/) ) = (/)
4	3	feq2i 6037	. . 3 |- ( (/) : ( (/) X. (/) ) --> RR <-> (/) : (/) --> RR )
5	2, 4	mpbir 221	. 2 |- (/) : ( (/) X. (/) ) --> RR
6		noel 3919	. . . 4 |- -. x e. (/)
7	6	pm2.21i 116	. . 3 |- ( x e. (/) -> ( ( x (/) y ) = 0 <-> x = y ) )
8	7	adantr 481	. 2 |- ( ( x e. (/) /\ y e. (/) ) -> ( ( x (/) y ) = 0 <-> x = y ) )
9	6	pm2.21i 116	. . 3 |- ( x e. (/) -> ( x (/) y ) <_ ( ( z (/) x ) + ( z (/) y ) ) )
10	9	3ad2ant1 1082	. 2 |- ( ( x e. (/) /\ y e. (/) /\ z e. (/) ) -> ( x (/) y ) <_ ( ( z (/) x ) + ( z (/) y ) ) )
11	1, 5, 8, 10	ismeti 22130	1 |- (/) e. ( Met ` (/) )

Syntax hints:   <-> wb 196   = wceq 1483   e. wcel 1990   (/)c0 3915   class class class wbr 4653   X. cxp 5112   -->wf 5884   ` cfv 5888  (class class class)co 6650   RRcr 9935   0cc0 9936   + caddc 9939   <_ cle 10075   Metcme 19732

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-map 7859  df-met 19740

This theorem is referenced by: (None)