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Theorem nelrdva 2862
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 2116 . 2  |-  ( (
ph  /\  B  e.  A )  ->  B  =  B )
2 eleq1 2178 . . . . . . 7  |-  ( x  =  B  ->  (
x  e.  A  <->  B  e.  A ) )
32anbi2d 457 . . . . . 6  |-  ( x  =  B  ->  (
( ph  /\  x  e.  A )  <->  ( ph  /\  B  e.  A ) ) )
4 neeq1 2296 . . . . . 6  |-  ( x  =  B  ->  (
x  =/=  B  <->  B  =/=  B ) )
53, 4imbi12d 233 . . . . 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 2718 . . . 4  |-  ( B  e.  A  ->  (
( ph  /\  B  e.  A )  ->  B  =/=  B ) )
87anabsi7 553 . . 3  |-  ( (
ph  /\  B  e.  A )  ->  B  =/=  B )
98neneqd 2304 . 2  |-  ( (
ph  /\  B  e.  A )  ->  -.  B  =  B )
101, 9pm2.65da 633 1  |-  ( ph  ->  -.  B  e.  A
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    = wceq 1314    e. wcel 1463    =/= wne 2283
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-v 2660
This theorem is referenced by: (None)
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