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Theorem co02 5374
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 5359 . 2  |-  Rel  ( A  o.  (/) )
2 rel0 4990 . 2  |-  Rel  (/)
3 noel 3624 . . . . . . 7  |-  -.  <. x ,  z >.  e.  (/)
4 df-br 4205 . . . . . . 7  |-  ( x
(/) z  <->  <. x ,  z >.  e.  (/) )
53, 4mtbir 291 . . . . . 6  |-  -.  x (/) z
65intnanr 882 . . . . 5  |-  -.  (
x (/) z  /\  z A y )
76nex 1564 . . . 4  |-  -.  E. z ( x (/) z  /\  z A y )
8 vex 2951 . . . . 5  |-  x  e. 
_V
9 vex 2951 . . . . 5  |-  y  e. 
_V
108, 9opelco 5035 . . . 4  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  E. z
( x (/) z  /\  z A y ) )
117, 10mtbir 291 . . 3  |-  -.  <. x ,  y >.  e.  ( A  o.  (/) )
12 noel 3624 . . 3  |-  -.  <. x ,  y >.  e.  (/)
1311, 122false 340 . 2  |-  ( <.
x ,  y >.  e.  ( A  o.  (/) )  <->  <. x ,  y >.  e.  (/) )
141, 2, 13eqrelriiv 4961 1  |-  ( A  o.  (/) )  =  (/)
Colors of variables: wff set class
Syntax hints:    /\ wa 359   E.wex 1550    = wceq 1652    e. wcel 1725   (/)c0 3620   <.cop 3809   class class class wbr 4204    o. ccom 4873
This theorem is referenced by:  co01  5375  gsumwmhm  14778  frmdgsum  14795  frmdup1  14797  efginvrel2  15347  0frgp  15399  ust0  18237  utop2nei  18268  tngds  18677  evl1fval  19935  dfpo2  25367
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2284  df-mo 2285  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-rab 2706  df-v 2950  df-dif 3315  df-un 3317  df-in 3319  df-ss 3326  df-nul 3621  df-if 3732  df-sn 3812  df-pr 3813  df-op 3815  df-br 4205  df-opab 4259  df-xp 4875  df-rel 4876  df-co 4878
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