Theorem nelprd 3731

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 2507	. . 3 |- ( ( A =/= B /\ A =/= C ) <-> -. ( A = B \/ A = C ) )
4		elpri 3728	. . . 4 |- ( A e. { B , C } -> ( A = B \/ A = C ) )
5	4	con3i 641	. . 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 415	1 |- ( ph -> -. A e. { B , C } )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   \/ wo 720   = wceq 1402   e. wcel 2209   =/= wne 2420   {cpr 3706

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-v 2823  df-un 3224  df-sn 3711  df-pr 3712

This theorem is referenced by:  tpfidisj  7226  sumtp  12159  perfectlem2  16028  eupth2lem3lem6fi  16626