Theorem nelprd 3553

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 2395	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3550	. . . 4 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
5	4	con3i 621	. . 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 408	1 |- ( ph -> -. A e. { B , C } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   =/= wne 2308   {cpr 3528

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534

This theorem is referenced by:  tpfidisj  6816  sumtp  11186