Theorem nelpri 3600

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair. (Contributed by David A. Wheeler, 10-May-2015.)

Hypotheses

Ref	Expression
nelpri.1	|- A =/= B
nelpri.2	|- A =/= C

Assertion

Ref	Expression
nelpri	|- -. A e. { B , C }

Proof of Theorem nelpri

Step	Hyp	Ref	Expression
1		nelpri.1	. 2 |- A =/= B
2		nelpri.2	. 2 |- 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	mp2an 423	1 |- -. A e. { B , C }

Syntax hints:   -. wn 3   /\ 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:  prneli  3601  pw1nel3  7187  sucpw1nel3  7189