Theorem prneli 3614

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 3613	. 2 |- -. A e. { B , C }
4	3	nelir 2443	1 |- A e/ { B , C }

Syntax hints:   =/= wne 2345   e/ wnel 2440   {cpr 3590

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 614  ax-in2 615  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-v 2737  df-un 3131  df-sn 3595  df-pr 3596

This theorem is referenced by: (None)