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Theorem co02 5186
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 5171 . 2  |-  Rel  ( A  o.  (/) )
2 rel0 4810 . 2  |-  Rel  (/)
3 noel 3459 . . . . . . 7  |-  -.  <. x ,  z >.  e.  (/)
4 df-br 4024 . . . . . . 7  |-  ( x
(/) z  <->  <. x ,  z >.  e.  (/) )
53, 4mtbir 290 . . . . . 6  |-  -.  x (/) z
65intnanr 881 . . . . 5  |-  -.  (
x (/) z  /\  z A y )
76nex 1542 . . . 4  |-  -.  E. z ( x (/) z  /\  z A y )
8 vex 2791 . . . . 5  |-  x  e. 
_V
9 vex 2791 . . . . 5  |-  y  e. 
_V
108, 9opelco 4853 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  E. z
( x (/) z  /\  z A y ) )
117, 10mtbir 290 . . 3  |-  -.  <. x ,  y >.  e.  ( A  o.  (/) )
12 noel 3459 . . 3  |-  -.  <. x ,  y >.  e.  (/)
1311, 122false 339 . 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 358   E.wex 1528    = wceq 1623    e. wcel 1684   (/)c0 3455   <.cop 3643   class class class wbr 4023    o. ccom 4693
This theorem is referenced by:  co01  5187  gsumwmhm  14467  frmdgsum  14484  frmdup1  14486  efginvrel2  15036  0frgp  15088  tngds  18164  evl1fval  19410  dfpo2  24112
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-br 4024  df-opab 4078  df-xp 4695  df-rel 4696  df-co 4698
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