Theorem eltpi 3622

Description: A member of an unordered triple of classes is one of them. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
eltpi	|- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) )

Proof of Theorem eltpi

Step	Hyp	Ref	Expression
1		eltpg 3620	. 2 |- ( A e. { B , C , D } -> ( A e. { B , C , D } <-> ( A = B \/ A = C \/ A = D ) ) )
2	1	ibi 175	1 |- ( A e. { B , C , D } -> ( A = B \/ A = C \/ A = D ) )

Syntax hints:   -> wi 4   \/ w3o 967   = wceq 1343   e. wcel 2136   {ctp 3577

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-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-3or 969  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-v 2727  df-un 3119  df-sn 3581  df-pr 3582  df-tp 3583

This theorem is referenced by:  prm23lt5  12191  zabsle1  13500