Theorem prneli 3608

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 3607	. 2 |- -. A e. { B , C }
4	3	nelir 2438	1 |- A e/ { B , C }

Syntax hints:   =/= wne 2340   e/ wnel 2435   {cpr 3584

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590

This theorem is referenced by: (None)