Theorem nelpri 3546

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 2393	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3545	. . . 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	mp2an 422	1 |- -. A e. { B , C }

Syntax hints:   -. wn 3   /\ wa 103   \/ wo 697   = wceq 1331   e. wcel 1480   =/= wne 2306   {cpr 3523

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 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-v 2683  df-un 3070  df-sn 3528  df-pr 3529

This theorem is referenced by:  prneli  3547