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Theorem co02 5185
Description: Composition with the empty set. Theorem 20 of [Suppes] p. 63. (Contributed by NM, 24-Apr-2004.)
Assertion
Ref Expression
co02  |-  ( A  o.  (/) )  =  (/)
Dummy variables  x  y  z are mutually distinct and distinct from all other variables.

Proof of Theorem co02
StepHypRef Expression
1 relco 5170 . 2  |-  Rel  ( A  o.  (/) )
2 rel0 4810 . 2  |-  Rel  (/)
3 noel 3461 . . . . . . 7  |-  -.  <. x ,  z >.  e.  (/)
4 df-br 4026 . . . . . . 7  |-  ( x
(/) z  <->  <. x ,  z >.  e.  (/) )
53, 4mtbir 292 . . . . . 6  |-  -.  x (/) z
65intnanr 883 . . . . 5  |-  -.  (
x (/) z  /\  z A y )
76nex 1543 . . . 4  |-  -.  E. z ( x (/) z  /\  z A y )
8 vex 2793 . . . . 5  |-  x  e. 
_V
9 vex 2793 . . . . 5  |-  y  e. 
_V
108, 9opelco 4853 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  E. z
( x (/) z  /\  z A y ) )
117, 10mtbir 292 . . 3  |-  -.  <. x ,  y >.  e.  ( A  o.  (/) )
12 noel 3461 . . 3  |-  -.  <. x ,  y >.  e.  (/)
1311, 122false 341 . 2  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  <. x ,  y >.  e.  (/) )
141, 2, 13eqrelriiv 4781 1  |-  ( A  o.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 360   E.wex 1529    = wceq 1624    e. wcel 1685   (/)c0 3457   <.cop 3645   class class class wbr 4025    o. ccom 4693
This theorem is referenced by:  co01  5186  gsumwmhm  14462  frmdgsum  14479  frmdup1  14481  efginvrel2  15031  0frgp  15083  tngds  18159  evl1fval  19405  dfpo2  23516
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 938  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-eu 2149  df-mo 2150  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-rab 2554  df-v 2792  df-dif 3157  df-un 3159  df-in 3161  df-ss 3168  df-nul 3458  df-if 3568  df-sn 3648  df-pr 3649  df-op 3651  df-br 4026  df-opab 4080  df-xp 4695  df-rel 4696  df-co 4698
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