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Theorem co02 5202
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 5187 . 2  |-  Rel  ( A  o.  (/) )
2 rel0 4826 . 2  |-  Rel  (/)
3 noel 3472 . . . . . . 7  |-  -.  <. x ,  z >.  e.  (/)
4 df-br 4040 . . . . . . 7  |-  ( x
(/) z  <->  <. x ,  z >.  e.  (/) )
53, 4mtbir 290 . . . . . 6  |-  -.  x (/) z
65intnanr 881 . . . . 5  |-  -.  (
x (/) z  /\  z A y )
76nex 1545 . . . 4  |-  -.  E. z ( x (/) z  /\  z A y )
8 vex 2804 . . . . 5  |-  x  e. 
_V
9 vex 2804 . . . . 5  |-  y  e. 
_V
108, 9opelco 4869 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  E. z
( x (/) z  /\  z A y ) )
117, 10mtbir 290 . . 3  |-  -.  <. x ,  y >.  e.  ( A  o.  (/) )
12 noel 3472 . . 3  |-  -.  <. x ,  y >.  e.  (/)
1311, 122false 339 . 2  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  <. x ,  y >.  e.  (/) )
141, 2, 13eqrelriiv 4797 1  |-  ( A  o.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 358   E.wex 1531    = wceq 1632    e. wcel 1696   (/)c0 3468   <.cop 3656   class class class wbr 4039    o. ccom 4709
This theorem is referenced by:  co01  5203  gsumwmhm  14483  frmdgsum  14500  frmdup1  14502  efginvrel2  15052  0frgp  15104  tngds  18180  evl1fval  19426  dfpo2  24183
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-eu 2160  df-mo 2161  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-ral 2561  df-rex 2562  df-rab 2565  df-v 2803  df-dif 3168  df-un 3170  df-in 3172  df-ss 3179  df-nul 3469  df-if 3579  df-sn 3659  df-pr 3660  df-op 3662  df-br 4040  df-opab 4094  df-xp 4711  df-rel 4712  df-co 4714
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