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Theorem nelrdva 2822
Description: Deduce negative membership from an implication. (Contributed by Thierry Arnoux, 27-Nov-2017.)
Hypothesis
Ref Expression
nelrdva.1  |-  ( (
ph  /\  x  e.  A )  ->  x  =/=  B )
Assertion
Ref Expression
nelrdva  |-  ( ph  ->  -.  B  e.  A
)
Distinct variable groups:    x, A    x, B    ph, x

Proof of Theorem nelrdva
StepHypRef Expression
1 eqidd 2089 . 2  |-  ( (
ph  /\  B  e.  A )  ->  B  =  B )
2 eleq1 2150 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
32anbi2d 452 . . . . . 6  |-  ( x  =  B  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  B  e.  A ) ) )
4 neeq1 2268 . . . . . 6  |-  ( x  =  B  ->  (
x  =/=  B  <->  B  =/=  B ) )
53, 4imbi12d 232 . . . . 5  |-  ( x  =  B  ->  (
( ( ph  /\  x  e.  A )  ->  x  =/=  B )  <-> 
( ( ph  /\  B  e.  A )  ->  B  =/=  B ) ) )
6 nelrdva.1 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  =/=  B )
75, 6vtoclg 2679 . . . 4  |-  ( B  e.  A  ->  (
( ph  /\  B  e.  A )  ->  B  =/=  B ) )
87anabsi7 548 . . 3  |-  ( (
ph  /\  B  e.  A )  ->  B  =/=  B )
98neneqd 2276 . 2  |-  ( (
ph  /\  B  e.  A )  ->  -.  B  =  B )
101, 9pm2.65da 622 1  |-  ( ph  ->  -.  B  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1289    e. wcel 1438    =/= wne 2255
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-ne 2256  df-v 2621
This theorem is referenced by: (None)
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