Theorem prneli 3499

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

Hypotheses

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

Assertion

Ref	Expression
prneli	|- A e/ { B , C }

Proof of Theorem prneli

Step	Hyp	Ref	Expression
1		prneli.1	. . 3 |- A =/= B
2		prneli.2	. . 3 |- A =/= C
3	1, 2	nelpri 3498	. 2 |- -. A e. { B , C }
4	3	nelir 2365	1 |- A e/ { B , C }

Syntax hints:   =/= wne 2267   e/ wnel 2362   {cpr 3475

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 584  ax-in2 585  ax-io 671  ax-5 1391  ax-7 1392  ax-gen 1393  ax-ie1 1437  ax-ie2 1438  ax-8 1450  ax-10 1451  ax-11 1452  ax-i12 1453  ax-bndl 1454  ax-4 1455  ax-17 1474  ax-i9 1478  ax-ial 1482  ax-i5r 1483  ax-ext 2082

This theorem depends on definitions:  df-bi 116  df-tru 1302  df-nf 1405  df-sb 1704  df-clab 2087  df-cleq 2093  df-clel 2096  df-nfc 2229  df-ne 2268  df-nel 2363  df-v 2643  df-un 3025  df-sn 3480  df-pr 3481

This theorem is referenced by: (None)