Theorem nelpri 3693

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 2489	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3692	. . . 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 715   = wceq 1397   e. wcel 2202   =/= wne 2402   {cpr 3670

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676

This theorem is referenced by:  prneli  3694  pw1nel3  7448  sucpw1nel3  7450