Theorem nelprd 3602

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)

Hypotheses

Ref	Expression
nelprd.1	|- ( ph -> A =/= B )
nelprd.2	|- ( ph -> A =/= C )

Assertion

Ref	Expression
nelprd	|- ( ph -> -. A e. { B , C } )

Proof of Theorem nelprd

Step	Hyp	Ref	Expression
1		nelprd.1	. 2 |- ( ph -> A =/= B )
2		nelprd.2	. 2 |- ( ph -> A =/= C )
3		neanior 2423	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3599	. . . 4 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
5	4	con3i 622	. . 3 |- ( -. ( A = B \/ A = C ) -> -. A e. { B , C } )
6	3, 5	sylbi 120	. 2 |- ( ( A =/= B /\ A =/= C ) -> -. A e. { B , C } )
7	1, 2, 6	syl2anc 409	1 |- ( ph -> -. A e. { B , C } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 698   = wceq 1343   e. wcel 2136   =/= wne 2336   {cpr 3577

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-v 2728  df-un 3120  df-sn 3582  df-pr 3583

This theorem is referenced by:  tpfidisj  6893  sumtp  11355