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Theorem eq0rdv 3402
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 608 . . 3  |-  ( ph  ->  ( x  e.  A  ->  x  e.  (/) ) )
32ssrdv 3098 . 2  |-  ( ph  ->  A  C_  (/) )
4 ss0 3398 . 2  |-  ( A 
C_  (/)  ->  A  =  (/) )
53, 4syl 14 1  |-  ( ph  ->  A  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    = wceq 1331    e. wcel 1480    C_ wss 3066   (/)c0 3358
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119
This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359
This theorem is referenced by:  exmid01  4116  dcextest  4490  nfvres  5447  map0b  6574  snon0  6817  snexxph  6831  fodju0  7012  fzdisj  9825  bldisj  12559
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