Theorem co02 5649

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.

Step	Hyp	Ref	Expression
1		relco 5633	. 2 |- Rel ( A o. (/) )
2		rel0 5243	. 2 |- Rel (/)
3		br0 4701	. . . . . 6 |- -. x (/) z
4	3	intnanr 961	. . . . 5 |- -. ( x (/) z /\ z A y )
5	4	nex 1731	. . . 4 |- -. E. z ( x (/) z /\ z A y )
6		vex 3203	. . . . 5 |- x e. _V
7		vex 3203	. . . . 5 |- y e. _V
8	6, 7	opelco 5293	. . . 4 |- ( <. x , y >. e. ( A o. (/) ) <-> E. z ( x (/) z /\ z A y ) )
9	5, 8	mtbir 313	. . 3 |- -. <. x , y >. e. ( A o. (/) )
10		noel 3919	. . 3 |- -. <. x , y >. e. (/)
11	9, 10	2false 365	. 2 |- ( <. x , y >. e. ( A o. (/) ) <-> <. x , y >. e. (/) )
12	1, 2, 11	eqrelriiv 5214	1 |- ( A o. (/) ) = (/)

Syntax hints:   /\ wa 384   = wceq 1483   E.wex 1704   e. wcel 1990   (/)c0 3915   <.cop 4183   class class class wbr 4653   o. ccom 5118

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-co 5123

This theorem is referenced by:  co01  5650  gsumwmhm  17382  frmdgsum  17399  frmdup1  17401  efginvrel2  18140  0frgp  18192  evl1fval  19692  utop2nei  22054  tngds  22452  mrsub0  31413  dfpo2  31645  cononrel1  37900