MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eq0rdv Structured version   Unicode version

Theorem eq0rdv 3662
Description: Deduction rule for equality to the empty set. (Contributed by NM, 11-Jul-2014.)
Hypothesis
Ref Expression
eq0rdv.1  |-  ( ph  ->  -.  x  e.  A
)
Assertion
Ref Expression
eq0rdv  |-  ( ph  ->  A  =  (/) )
Distinct variable groups:    x, A    ph, x

Proof of Theorem eq0rdv
StepHypRef Expression
1 eq0rdv.1 . . . 4  |-  ( ph  ->  -.  x  e.  A
)
21pm2.21d 100 . . 3  |-  ( ph  ->  ( x  e.  A  ->  x  e.  (/) ) )
32ssrdv 3354 . 2  |-  ( ph  ->  A  C_  (/) )
4 ss0 3658 . 2  |-  ( A 
C_  (/)  ->  A  =  (/) )
53, 4syl 16 1  |-  ( ph  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1652    e. wcel 1725    C_ wss 3320   (/)c0 3628
This theorem is referenced by:  map0b  7052  disjen  7264  mapdom1  7272  pwxpndom2  8540  fzdisj  11078  smu01lem  12997  prmreclem5  13288  vdwap0  13344  natfval  14143  fucbas  14157  fuchom  14158  coafval  14219  efgval  15349  lsppratlem6  16224  lbsextlem4  16233  psrvscafval  16454  cfinufil  17960  ufinffr  17961  fin1aufil  17964  bldisj  18428  reconnlem1  18857  pcofval  19035  bcthlem5  19281  volfiniun  19441  fta1g  20090  fta1  20225  rpvmasum  21220  ipo0  27628  ifr0  27629
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-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-v 2958  df-dif 3323  df-in 3327  df-ss 3334  df-nul 3629
  Copyright terms: Public domain W3C validator