Theorem prneli 3693

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 3692	. 2 |- -. A e. { B , C }
4	3	nelir 2499	1 |- A e/ { B , C }

Syntax hints:   =/= wne 2401   e/ wnel 2496   {cpr 3669

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 2212

This theorem depends on definitions:  df-bi 117  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ne 2402  df-nel 2497  df-v 2803  df-un 3203  df-sn 3674  df-pr 3675

This theorem is referenced by: (None)