MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  co02 Structured version   Unicode version

Theorem co02 5520
Description: Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
co02  |-  ( A  o.  (/) )  =  (/)

Proof of Theorem co02
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relco 5504 . 2  |-  Rel  ( A  o.  (/) )
2 rel0 5126 . 2  |-  Rel  (/)
3 noel 3789 . . . . . . 7  |-  -.  <. x ,  z >.  e.  (/)
4 df-br 4448 . . . . . . 7  |-  ( x
(/) z  <->  <. x ,  z >.  e.  (/) )
53, 4mtbir 299 . . . . . 6  |-  -.  x (/) z
65intnanr 913 . . . . 5  |-  -.  (
x (/) z  /\  z A y )
76nex 1610 . . . 4  |-  -.  E. z ( x (/) z  /\  z A y )
8 vex 3116 . . . . 5  |-  x  e. 
_V
9 vex 3116 . . . . 5  |-  y  e. 
_V
108, 9opelco 5173 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  E. z
( x (/) z  /\  z A y ) )
117, 10mtbir 299 . . 3  |-  -.  <. x ,  y >.  e.  ( A  o.  (/) )
12 noel 3789 . . 3  |-  -.  <. x ,  y >.  e.  (/)
1311, 122false 350 . 2  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  <. x ,  y >.  e.  (/) )
141, 2, 13eqrelriiv 5096 1  |-  ( A  o.  (/) )  =  (/)
Colors of variables: wff setvar class
Syntax hints:    /\ wa 369    = wceq 1379   E.wex 1596    e. wcel 1767   (/)c0 3785   <.cop 4033   class class class wbr 4447    o. ccom 5003
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-br 4448  df-opab 4506  df-xp 5005  df-rel 5006  df-co 5008
This theorem is referenced by:  co01  5521  gsumwmhm  15842  frmdgsum  15859  frmdup1  15861  efginvrel2  16548  0frgp  16600  evl1fval  18151  ust0  20473  utop2nei  20504  tngds  20913  dfpo2  28777
  Copyright terms: Public domain W3C validator