Theorem nelprd 3699

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 2490	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3696	. . . 4 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
5	4	con3i 637	. . 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 716   = wceq 1398   e. wcel 2202   =/= wne 2403   {cpr 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ne 2404  df-v 2805  df-un 3205  df-sn 3679  df-pr 3680

This theorem is referenced by:  tpfidisj  7164  sumtp  12055  perfectlem2  15814  eupth2lem3lem6fi  16412