Theorem eltpi 3630

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 3628	. 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 972   = wceq 1348   e. wcel 2141   {ctp 3585

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3or 974  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-sn 3589  df-pr 3590  df-tp 3591

This theorem is referenced by:  prm23lt5  12217  zabsle1  13694