Theorem nelpri 3718

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 2501	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3717	. . . 4 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
5	4	con3i 637	. . 3 |- ( -. ( A = B \/ A = C ) -> -. A e. { B , C } )
6	3, 5	sylbi 121	. 2 |- ( ( A =/= B /\ A =/= C ) -> -. A e. { B , C } )
7	1, 2, 6	mp2an 426	1 |- -. A e. { B , C }

Syntax hints:   -. wn 3   /\ wa 104   \/ wo 716   = wceq 1398   e. wcel 2205   =/= wne 2414   {cpr 3695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-v 2817  df-un 3218  df-sn 3700  df-pr 3701

This theorem is referenced by:  prneli  3719  pw1nel3  7554  sucpw1nel3  7556