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Theorem dom0 7638
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 7515 . . . . 5  |-  Rel  ~<_
21brrelexi 5029 . . . 4  |-  ( A  ~<_  (/)  ->  A  e.  _V )
3 0domg 7637 . . . 4  |-  ( A  e.  _V  ->  (/)  ~<_  A )
42, 3syl 16 . . 3  |-  ( A  ~<_  (/)  ->  (/)  ~<_  A )
54pm4.71i 630 . 2  |-  ( A  ~<_  (/) 
<->  ( A  ~<_  (/)  /\  (/)  ~<_  A ) )
6 sbthb 7631 . 2  |-  ( ( A  ~<_  (/)  /\  (/)  ~<_  A )  <-> 
A  ~~  (/) )
7 en0 7571 . 2  |-  ( A 
~~  (/)  <->  A  =  (/) )
85, 6, 73bitri 271 1  |-  ( A  ~<_  (/) 
<->  A  =  (/) )
Colors of variables: wff setvar class
Syntax hints:    <-> wb 184    /\ wa 367    = wceq 1398    e. wcel 1823   _Vcvv 3106   (/)c0 3783   class class class wbr 4439    ~~ cen 7506    ~<_ cdom 7507
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pow 4615  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-rab 2813  df-v 3108  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-pw 4001  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-er 7303  df-en 7510  df-dom 7511
This theorem is referenced by:  pwcdadom  8587  fin1a2lem11  8781  cfpwsdom  8950
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