MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  dom0 Structured version   Unicode version

Theorem dom0 7642
Description: A set dominated by the empty set is empty. (Contributed by NM, 22-Nov-2004.)
Assertion
Ref Expression
dom0  |-  ( A  ~<_  (/) 
<->  A  =  (/) )

Proof of Theorem dom0
StepHypRef Expression
1 reldom 7519 . . . . 5  |-  Rel  ~<_
21brrelexi 5039 . . . 4  |-  ( A  ~<_  (/)  ->  A  e.  _V )
3 0domg 7641 . . . 4  |-  ( A  e.  _V  ->  (/)  ~<_  A )
42, 3syl 16 . . 3  |-  ( A  ~<_  (/)  ->  (/)  ~<_  A )
54pm4.71i 632 . 2  |-  ( A  ~<_  (/) 
<->  ( A  ~<_  (/)  /\  (/)  ~<_  A ) )
6 sbthb 7635 . 2  |-  ( ( A  ~<_  (/)  /\  (/)  ~<_  A )  <-> 
A  ~~  (/) )
7 en0 7575 . 2  |-  ( A 
~~  (/)  <->  A  =  (/) )
85, 6, 73bitri 271 1  |-  ( A  ~<_  (/) 
<->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 369    = wceq 1379    e. wcel 1767   _Vcvv 3113   (/)c0 3785   class class class wbr 4447    ~~ cen 7510    ~<_ cdom 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6574
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-er 7308  df-en 7514  df-dom 7515
This theorem is referenced by:  pwcdadom  8592  fin1a2lem11  8786  cfpwsdom  8955
  Copyright terms: Public domain W3C validator