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Theorem eq0rdv 3327
Description: Deduction 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 584 . . 3  |-  ( ph  ->  ( x  e.  A  ->  x  e.  (/) ) )
32ssrdv 3031 . 2  |-  ( ph  ->  A  C_  (/) )
4 ss0 3323 . 2  |-  ( A 
C_  (/)  ->  A  =  (/) )
53, 4syl 14 1  |-  ( ph  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1289    e. wcel 1438    C_ wss 2999   (/)c0 3286
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-v 2621  df-dif 3001  df-in 3005  df-ss 3012  df-nul 3287
This theorem is referenced by:  exmid01  4032  dcextest  4396  nfvres  5337  map0b  6442  snon0  6643  snexxph  6657  fodjuomnilem0  6800  fzdisj  9464
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