Theorem nelprd 3669

Description: If an element doesn't match the items in an unordered pair, it is not in the unordered pair, deduction version. (Contributed by Alexander van der Vekens, 25-Jan-2018.)

Hypotheses

Ref	Expression
nelprd.1	|- ( ph -> A =/= B )
nelprd.2	|- ( ph -> A =/= C )

Assertion

Ref	Expression
nelprd	|- ( ph -> -. A e. { B , C } )

Proof of Theorem nelprd

Step	Hyp	Ref	Expression
1		nelprd.1	. 2 |- ( ph -> A =/= B )
2		nelprd.2	. 2 |- ( ph -> A =/= C )
3		neanior 2465	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3666	. . . 4 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
5	4	con3i 633	. . 3 |- ( -. ( A = B \/ A = C ) -> -. A e. { B , C } )
6	3, 5	sylbi 121	. 2 |- ( ( A =/= B /\ A =/= C ) -> -. A e. { B , C } )
7	1, 2, 6	syl2anc 411	1 |- ( ph -> -. A e. { B , C } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 710   = wceq 1373   e. wcel 2178   =/= wne 2378   {cpr 3644

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 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650

This theorem is referenced by:  tpfidisj  7052  sumtp  11840  perfectlem2  15587