Theorem nelpri 3755

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

Hypotheses

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

Assertion

Ref	Expression
nelpri	|- -. A e. {B, C}

Proof of Theorem nelpri

Step	Hyp	Ref	Expression
1		nelpri.1	. 2 |- A =/= B
2		nelpri.2	. 2 |- A =/= C
3		neanior 2602	. . 3 |- ((A =/= B /\ A =/= C) <-> -. (A = B \/ A = C))
4		elpri 3754	. . . 4 |- (A e. {B, C} -> (A = B \/ A = C))
5	4	con3i 127	. . 3 |- (-. (A = B \/ A = C) -> -. A e. {B, C})
6	3, 5	sylbi 187	. 2 |- ((A =/= B /\ A =/= C) -> -. A e. {B, C})
7	1, 2, 6	mp2an 653	1 |- -. A e. {B, C}

Syntax hints:  -. wn 3   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710   =/= wne 2517  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-sn 3742  df-pr 3743

This theorem is referenced by: (None)