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Theorem nelrdva 2807
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 2084 . 2  |-  ( (
ph  /\  B  e.  A )  ->  B  =  B )
2 eleq1 2145 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
32anbi2d 452 . . . . . 6  |-  ( x  =  B  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  B  e.  A ) ) )
4 neeq1 2262 . . . . . 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 2667 . . . 4  |-  ( B  e.  A  ->  (
( ph  /\  B  e.  A )  ->  B  =/=  B ) )
87anabsi7 546 . . 3  |-  ( (
ph  /\  B  e.  A )  ->  B  =/=  B )
98neneqd 2270 . 2  |-  ( (
ph  /\  B  e.  A )  ->  -.  B  =  B )
101, 9pm2.65da 620 1  |-  ( ph  ->  -.  B  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 102    = wceq 1285    e. wcel 1434    =/= wne 2249
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 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-v 2612
This theorem is referenced by: (None)
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