Theorem prneli 3595

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 3594	. 2 |- -. A e. { B , C }
4	3	nelir 2432	1 |- A e/ { B , C }

Syntax hints:   =/= wne 2334   e/ wnel 2429   {cpr 3571

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 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146

This theorem depends on definitions:  df-bi 116  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-v 2723  df-un 3115  df-sn 3576  df-pr 3577

This theorem is referenced by: (None)